Abstract:
Disclosed are a row-vector norm comparison method and a row-vector norm comparison apparatus for an inverse matrix. A row-vector norm comparison apparatus includes: an input matrix processing module that receives and combines constituent elements of a matrix; a cofactor operation module that multiplexes the combination result of the constituent elements to calculate factors constituting an adjoint matrix; a square calculation module that squares the calculated factors; a summation module that selects a predetermined number of factors among the squared factors and sums the selected factors to calculate the norms of row vectors in an inverse matrix; and a norm comparison module that outputs a comparison result of the calculated norms of the row vectors. A row-vector norm comparison method includes: combining constituent elements of a matrix to generates a plurality of combination results of the constituent elements; multiplexing the combination results to calculate factors constituting an adjoint matrix of the matrix; squaring the calculated factors and selectively summing the squared factors; and calculating the norms of row vectors in an inverse matrix and comparing the calculated norms of the row vectors. With this configuration, row-vector norm comparison for an inverse matrix can be performed with a design structure. Therefore, it is not necessary to use an existing complex operation method. In addition, low power consumption of the multiple antenna system can be achieved, and efficiency of the design structure can be improved.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims priority to and the benefit of Korean Patent Application No. 10-2007-0133663 filed in the Korean Intellectual Property Office on Dec. 18, 2007, the entire contents of which are incorporated herein by reference. 
     BACKGROUND OF THE INVENTION 
     (a) Field of the Invention 
     The present invention relates to a row-vector norm comparison method and a row-vector norm comparison apparatus for an inverse matrix. In particular, the present invention relates to a row-vector norm comparison method and a row-vector norm comparison apparatus for an inverse matrix of a 4×4 matrix for design of a multiple antenna system. 
     (b) Description of the Related Art 
     In a mobile communication system, a base station and a terminal transmit/receive a call with multiple antennas, thereby improving signal reliability. The multiple antenna system that transmits/receives a call with the multiple antenna may be designed by using a 4×4 matrix. 
     The multiple antenna system designed with the 4×4 matrix requires row-vector norm comparison for an inverse matrix in order to execute a detection algorithm for transmitted/received signal prediction. 
     To this end, a method that calculates an inverse matrix of the 4×4 matrix constituting the multiple antenna system, and performs the row-vector norm comparison for the inverse matrix based on the calculation result, has been used. 
     To calculate the inverse matrix of the 4×4 matrix requires many operations and the structure becomes complex, which makes it difficult to design a multiple antenna system. In order to solve this problem, various studies on technology and systems capable of easily calculating the inverse matrix of the 4×4 matrix with a simple operation structure have been undertaken. 
     However, there has been no suggested system that performs row-vector norm comparison for an inverse matrix directly from the 4×4 matrix constituting the multiple antenna system. 
     The above information disclosed in this Background section is only for enhancement of understanding of the background of the invention and therefore it may contain information that does not form the prior art that is already known in this country to a person of ordinary skill in the art. 
     SUMMARY OF THE INVENTION 
     The present invention has been made in an effort to provide a row-vector norm comparison apparatus and a row-vector norm comparison method for an inverse matrix that is capable of performing row-vector norm comparison for an inverse matrix directly from a 4×4 matrix used to design a multiple antenna system, thereby simplifying a design structure. 
     An exemplary embodiment of the present invention provides a row-vector norm comparison apparatus for an inverse matrix that performs row-vector norm comparison for an inverse matrix of a matrix. The row-vector norm comparison apparatus includes: an input matrix processing module that receives constituent elements of the matrix and combines the input constituent elements; a cofactor operation module that multiplexes the combination result of the constituent elements output from the input matrix processing module to calculate factors constituting an adjoint matrix of the matrix; a square calculation module that squares the factors output from the cofactor operation module; a summation module that selects a predetermined number of factors among the squared factors and sums the selected factors to calculate the norms of row-vectors in the inverse matrix; and a norm comparison module that compares the row-vector norms calculated by the summation module and outputs the comparison result. 
     Another embodiment of the present invention provides a row-vector norm comparison method for an inverse matrix that performs row-vector norm comparison for an inverse matrix of a matrix. The row-vector norm comparison method includes: receiving constituent elements of the matrix and combining the input constituent elements to generate a plurality of combination results of the constituent elements; multiplexing the plurality of combination results of the constituent elements to calculate factors constituting an adjoint matrix of the matrix; squaring the calculated factors and selectively summing the squared factors to calculate the norm of each row-vector of the inverse matrix; and comparing the calculated row-vector norms and outputting the comparison result. 
     According to the embodiments of the present invention, the multiple antenna system that requires row-vector norm comparison for an inverse matrix can be implemented with a simple design structure, without fully designing an inverse matrix operation structure that causes complexity and difficulty in design. Therefore, it is not necessary to use an existing complex operation method. In addition, low power consumption of the multiple antenna system can be achieved, and efficiency of the design structure can be improved. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram schematically illustrating a row-vector norm comparison apparatus according to an exemplary embodiment of the present invention. 
         FIG. 2  is a diagram schematically illustrating the internal configuration for explaining a function of an input matrix processing module according to an exemplary embodiment of the present invention. 
         FIG. 3  is a diagram schematically illustrating the internal configuration of a first cofactor operation module according to an exemplary embodiment of the present invention. 
         FIG. 4  is a diagram schematically illustrating the internal configuration of a common factor generation module according to an exemplary embodiment of the present invention. 
         FIG. 5  is shows diagrams illustrating the internal configurations of first common factor generation modules for calculating cofactors c 11  to c 14  adapted in various ways based on kinds of input factors according to an exemplary embodiment of the present invention. 
         FIG. 6  is a diagram schematically illustrating the internal configuration of a second cofactor operation module according to an exemplary embodiment of the present invention. 
         FIG. 7  is a diagram schematically illustrating the internal configuration of a norm comparison module according to an exemplary embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     In the following detailed description, only certain exemplary embodiments of the present invention have been shown and described, simply by way of illustration. As those skilled in the art would realize, the described embodiments may be modified in various different ways, all without departing from the spirit or scope of the present invention. Accordingly, the drawings and description are to be regarded as illustrative in nature and not restrictive. Like reference numerals designate like elements throughout the specification. 
     In addition, unless explicitly described to the contrary, the word “comprise” and variations such as “comprises” or “comprising” will be understood to imply the inclusion of stated elements but not the exclusion of any other elements. The terms “-er”, “-or”, and “module” described in the specification mean units for processing at least one function and operation and can be implemented by hardware components, and combinations of hardware components and software components. 
     Assuming a 4×4 input matrix according to an exemplary embodiment of the present invention is A, the matrix A may be expressed by Equation 1. 
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     [ 
                     
                       
                         
                           a 
                         
                         
                           b 
                         
                         
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                           p 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     An inverse matrix A −1  of the matrix A having elements a to p may be expressed by Equation 2. 
     
       
         
           
             
               
                 
                   
                     A 
                     
                       - 
                       1 
                     
                   
                   = 
                   
                     
                       1 
                       
                         det 
                         ⁡ 
                         
                           ( 
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                           ) 
                         
                       
                     
                     ⁢ 
                     
                       adj 
                       ⁡ 
                       
                         ( 
                         A 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
     Here, adj(A) is an adjoint matrix of the matrix A, and det(A) is a determinant of the matrix A. The elements of the inverse matrix A −1  are all multiplied by 
               1     det   ⁡     (   A   )         .         
Therefore, there is no affect on row-vector norm comparison for an inverse matrix according to an exemplary embodiment of the present invention. For this reason, what is necessary for row-vector norm comparison for an inverse matrix according to an exemplary embodiment of the present invention is to calculate the adjoint matrix adj(A).
 
     The adjoint matrix adj(A) is a transpose matrix of a cofactor matrix c ij . Accordingly, if the cofactor matrix is calculated, it is possible to calculate a row vector of an inverse matrix. Here, the adjoint matrix adj(A) may be expressed by Equation 3. 
     
       
         
           
             
               
                 
                   
                     adj 
                     ⁡ 
                     
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                               11 
                             
                           
                         
                         
                           
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                               12 
                             
                           
                         
                         
                           
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                               c 
                               24 
                             
                           
                         
                         
                           
                             - 
                             
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                               34 
                             
                           
                         
                         
                           
                             + 
                             
                               c 
                               44 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     3 
                   
                   ) 
                 
               
             
           
         
       
     
     In the adjoint matrix adj(A), what is necessary is for row-vector norm comparison of an inverse matrix to compare the norms of the row vectors expressed by Equations 4 to 7.
 
Norm of First Row Vector= c   11   2   +c   21   2   +c   31   2   +c   41   2   (Equation 4)
 
Norm of Second Row Vector= c   12   2   +c   22   2   +c   32   2   +c   42   2   (Equation 5)
 
Norm of Third Row Vector= c   13   2   +c   23   2   +c   33   2   +c   43   2   (Equation 6)
 
Norm of Fourth Row Vector= c   14   2   +c   24   2   +c   34   2   +c   44   2   (Equation 7)
 
     That is, what is necessary for row-vector norm comparison for an inverse matrix according to an exemplary embodiment of the present invention is to calculate c 11   2 , c 12   2 , c 13   2 , c 14   2 , c 21   2 , c 22   2 , c 23   2 , c 24   2 , c 31   2 , c 32   2 , c 33   2 , c 34   2 , c 41   2 , c 42   2 , c 43   2 , and c 44   2 . 
     Here, cofactors constituting the adjoint matrix adj(A), that is, c 11  to c 44 , may be calculated by Equation 8. 
     
       
         
           
             
               
                 
                   
                     
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                                     n 
                                   
                                 
                               
                                
                             
                           
                         
                         = 
                         
                           
                             a 
                             ⁡ 
                             
                               ( 
                               
                                 jp 
                                 - 
                                 nl 
                               
                               ) 
                             
                           
                           - 
                           
                             b 
                             ⁡ 
                             
                               ( 
                               
                                 ip 
                                 - 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   l 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             d 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                                 - 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   m 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       c 
                       24 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               a 
                             
                             
                               b 
                             
                             
                               c 
                             
                           
                           
                             
                               i 
                             
                             
                               j 
                             
                             
                               k 
                             
                           
                           
                             
                               m 
                             
                             
                               n 
                             
                             
                               o 
                             
                           
                         
                          
                       
                       = 
                       
                         
                           
                             a 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     j 
                                   
                                   
                                     k 
                                   
                                 
                                 
                                   
                                     n 
                                   
                                   
                                     o 
                                   
                                 
                               
                                
                             
                           
                           - 
                           
                             b 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     i 
                                   
                                   
                                     k 
                                   
                                 
                                 
                                   
                                     m 
                                   
                                   
                                     o 
                                   
                                 
                               
                                
                             
                           
                           + 
                           
                             c 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     i 
                                   
                                   
                                     j 
                                   
                                 
                                 
                                   
                                     m 
                                   
                                   
                                     n 
                                   
                                 
                               
                                
                             
                           
                         
                         = 
                         
                           
                             a 
                             ⁡ 
                             
                               ( 
                               
                                 jo 
                                 - 
                                 nk 
                               
                               ) 
                             
                           
                           - 
                           
                             b 
                             ⁡ 
                             
                               ( 
                               
                                 io 
                                 - 
                                 mk 
                               
                               ) 
                             
                           
                           + 
                           
                             c 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                                 - 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   m 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       c 
                       31 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               b 
                             
                             
                               c 
                             
                             
                               d 
                             
                           
                           
                             
                               f 
                             
                             
                               g 
                             
                             
                               h 
                             
                           
                           
                             
                               n 
                             
                             
                               o 
                             
                             
                               p 
                             
                           
                         
                          
                       
                       = 
                       
                         
                           
                             b 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     g 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     o 
                                   
                                   
                                     p 
                                   
                                 
                               
                                
                             
                           
                           - 
                           
                             c 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     f 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     n 
                                   
                                   
                                     p 
                                   
                                 
                               
                                
                             
                           
                           + 
                           
                             d 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     f 
                                   
                                   
                                     g 
                                   
                                 
                                 
                                   
                                     n 
                                   
                                   
                                     o 
                                   
                                 
                               
                                
                             
                           
                         
                         = 
                         
                           
                             b 
                             ⁡ 
                             
                               ( 
                               
                                 gp 
                                 - 
                                 ho 
                               
                               ) 
                             
                           
                           - 
                           
                             c 
                             ⁡ 
                             
                               ( 
                               
                                 fp 
                                 - 
                                 nh 
                               
                               ) 
                             
                           
                           + 
                           
                             d 
                             ⁡ 
                             
                               ( 
                               
                                 fo 
                                 - 
                                 gn 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       c 
                       32 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               a 
                             
                             
                               c 
                             
                             
                               d 
                             
                           
                           
                             
                               e 
                             
                             
                               g 
                             
                             
                               h 
                             
                           
                           
                             
                               m 
                             
                             
                               o 
                             
                             
                               p 
                             
                           
                         
                          
                       
                       = 
                       
                         
                           
                             a 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     g 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     o 
                                   
                                   
                                     p 
                                   
                                 
                               
                                
                             
                           
                           - 
                           
                             c 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     m 
                                   
                                   
                                     p 
                                   
                                 
                               
                                
                             
                           
                           + 
                           
                             d 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     g 
                                   
                                 
                                 
                                   
                                     m 
                                   
                                   
                                     o 
                                   
                                 
                               
                                
                             
                           
                         
                         = 
                         
                           
                             a 
                             ⁡ 
                             
                               ( 
                               
                                 gp 
                                 - 
                                 ho 
                               
                               ) 
                             
                           
                           - 
                           
                             c 
                             ⁡ 
                             
                               ( 
                               
                                 ep 
                                 - 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   h 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             d 
                             ⁡ 
                             
                               ( 
                               
                                 eo 
                                 - 
                                 
                                   g 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   m 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       c 
                       33 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               a 
                             
                             
                               b 
                             
                             
                               d 
                             
                           
                           
                             
                               e 
                             
                             
                               f 
                             
                             
                               h 
                             
                           
                           
                             
                               m 
                             
                             
                               n 
                             
                             
                               p 
                             
                           
                         
                          
                       
                       = 
                       
                         
                           
                             a 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     f 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     n 
                                   
                                   
                                     p 
                                   
                                 
                               
                                
                             
                           
                           - 
                           
                             b 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     m 
                                   
                                   
                                     p 
                                   
                                 
                               
                                
                             
                           
                           + 
                           
                             d 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     f 
                                   
                                 
                                 
                                   
                                     m 
                                   
                                   
                                     n 
                                   
                                 
                               
                                
                             
                           
                         
                         = 
                         
                           
                             a 
                             ⁡ 
                             
                               ( 
                               
                                 fp 
                                 - 
                                 hn 
                               
                               ) 
                             
                           
                           - 
                           
                             b 
                             ⁡ 
                             
                               ( 
                               
                                 ep 
                                 - 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   h 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             d 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   e 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                                 - 
                                 
                                   f 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   m 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       c 
                       34 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               a 
                             
                             
                               b 
                             
                             
                               c 
                             
                           
                           
                             
                               e 
                             
                             
                               f 
                             
                             
                               g 
                             
                           
                           
                             
                               m 
                             
                             
                               n 
                             
                             
                               o 
                             
                           
                         
                          
                       
                       = 
                       
                         
                           
                             a 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     f 
                                   
                                   
                                     g 
                                   
                                 
                                 
                                   
                                     n 
                                   
                                   
                                     o 
                                   
                                 
                               
                                
                             
                           
                           - 
                           
                             b 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     g 
                                   
                                 
                                 
                                   
                                     m 
                                   
                                   
                                     o 
                                   
                                 
                               
                                
                             
                           
                           + 
                           
                             c 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     f 
                                   
                                 
                                 
                                   
                                     m 
                                   
                                   
                                     n 
                                   
                                 
                               
                                
                             
                           
                         
                         = 
                         
                           
                             a 
                             ⁡ 
                             
                               ( 
                               
                                 fo 
                                 - 
                                 gn 
                               
                               ) 
                             
                           
                           - 
                           
                             b 
                             ⁡ 
                             
                               ( 
                               
                                 eo 
                                 - 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   g 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             c 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   e 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                                 - 
                                 fm 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       c 
                       41 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               b 
                             
                             
                               c 
                             
                             
                               d 
                             
                           
                           
                             
                               f 
                             
                             
                               g 
                             
                             
                               h 
                             
                           
                           
                             
                               j 
                             
                             
                               k 
                             
                             
                               l 
                             
                           
                         
                          
                       
                       = 
                       
                         
                           
                             b 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     g 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     k 
                                   
                                   
                                     l 
                                   
                                 
                               
                                
                             
                           
                           - 
                           
                             c 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     f 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     j 
                                   
                                   
                                     l 
                                   
                                 
                               
                                
                             
                           
                           + 
                           
                             d 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     f 
                                   
                                   
                                     g 
                                   
                                 
                                 
                                   
                                     j 
                                   
                                   
                                     k 
                                   
                                 
                               
                                
                             
                           
                         
                         = 
                         
                           
                             b 
                             ⁡ 
                             
                               ( 
                               
                                 gl 
                                 - 
                                 hk 
                               
                               ) 
                             
                           
                           - 
                           
                             c 
                             ⁡ 
                             
                               ( 
                               
                                 fl 
                                 - 
                                 jh 
                               
                               ) 
                             
                           
                           + 
                           
                             d 
                             ⁡ 
                             
                               ( 
                               
                                 fk 
                                 - 
                                 gj 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       c 
                       42 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               a 
                             
                             
                               c 
                             
                             
                               d 
                             
                           
                           
                             
                               e 
                             
                             
                               g 
                             
                             
                               h 
                             
                           
                           
                             
                               i 
                             
                             
                               k 
                             
                             
                               l 
                             
                           
                         
                          
                       
                       = 
                       
                         
                           
                             a 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     g 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     k 
                                   
                                   
                                     l 
                                   
                                 
                               
                                
                             
                           
                           - 
                           
                             c 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     i 
                                   
                                   
                                     l 
                                   
                                 
                               
                                
                             
                           
                           + 
                           
                             d 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     g 
                                   
                                 
                                 
                                   
                                     i 
                                   
                                   
                                     k 
                                   
                                 
                               
                                
                             
                           
                         
                         = 
                         
                           
                             a 
                             ⁡ 
                             
                               ( 
                               
                                 gl 
                                 - 
                                 hk 
                               
                               ) 
                             
                           
                           - 
                           
                             c 
                             ⁡ 
                             
                               ( 
                               
                                 el 
                                 - 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   h 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             d 
                             ⁡ 
                             
                               ( 
                               
                                 ek 
                                 - 
                                 
                                   g 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   i 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       c 
                       43 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               a 
                             
                             
                               b 
                             
                             
                               d 
                             
                           
                           
                             
                               e 
                             
                             
                               f 
                             
                             
                               h 
                             
                           
                           
                             
                               i 
                             
                             
                               j 
                             
                             
                               l 
                             
                           
                         
                          
                       
                       = 
                       
                         
                           
                             a 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     f 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     j 
                                   
                                   
                                     l 
                                   
                                 
                               
                                
                             
                           
                           - 
                           
                             b 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     h 
                                   
                                 
                                 
                                   
                                     i 
                                   
                                   
                                     l 
                                   
                                 
                               
                                
                             
                           
                           + 
                           
                             d 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     f 
                                   
                                 
                                 
                                   
                                     i 
                                   
                                   
                                     j 
                                   
                                 
                               
                                
                             
                           
                         
                         = 
                         
                           
                             a 
                             ⁡ 
                             
                               ( 
                               
                                 fl 
                                 - 
                                 hj 
                               
                               ) 
                             
                           
                           - 
                           
                             b 
                             ⁡ 
                             
                               ( 
                               
                                 el 
                                 - 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   h 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             d 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   e 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   j 
                                 
                                 - 
                                 
                                   f 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   i 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       c 
                       44 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               a 
                             
                             
                               b 
                             
                             
                               c 
                             
                           
                           
                             
                               e 
                             
                             
                               f 
                             
                             
                               g 
                             
                           
                           
                             
                               i 
                             
                             
                               j 
                             
                             
                               k 
                             
                           
                         
                          
                       
                       = 
                       
                         
                           
                             a 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     f 
                                   
                                   
                                     g 
                                   
                                 
                                 
                                   
                                     j 
                                   
                                   
                                     k 
                                   
                                 
                               
                                
                             
                           
                           - 
                           
                             b 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     g 
                                   
                                 
                                 
                                   
                                     i 
                                   
                                   
                                     k 
                                   
                                 
                               
                                
                             
                           
                           + 
                           
                             c 
                             ⁢ 
                             
                                
                               
                                 
                                   
                                     e 
                                   
                                   
                                     f 
                                   
                                 
                                 
                                   
                                     i 
                                   
                                   
                                     j 
                                   
                                 
                               
                                
                             
                           
                         
                         = 
                         
                           
                             a 
                             ⁡ 
                             
                               ( 
                               
                                 fk 
                                 - 
                                 gj 
                               
                               ) 
                             
                           
                           - 
                           
                             b 
                             ⁡ 
                             
                               ( 
                               
                                 ek 
                                 - 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   g 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             c 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   e 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   j 
                                 
                                 - 
                                 fi 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
     As expressed by Equation 8, the cofactors constituting the adjoint matrix adj(A) may be expressed by the elements of the matrix A. An exemplary embodiment of the present invention provides a row-vector norm comparison apparatus that calculates the cofactors constituting the adjoint matrix adj(A) and compares the norms of the row vectors expressed by Equations 4 to 7 on the basis of the calculated cofactors. 
       FIG. 1  is a block diagram schematically illustrating a row-vector norm comparison apparatus according to an exemplary embodiment of the present invention. 
     A row-vector norm comparison apparatus according to an exemplary embodiment of the present invention includes an input matrix processing module  110 , a first cofactor operation module  120 , a second cofactor operation module  130 , a square calculation module  140 , a summation module  150 , and a norm comparison module  160 . 
     The input matrix processing module  110  transmits input constituent elements of a matrix to the first cofactor operation module  120  and the second cofactor operation module  130 . The function of the input matrix processing module  110  will be described with reference to  FIG. 2 . 
       FIG. 2  is a diagram schematically illustrating the internal configuration for explaining a function of an input matrix processing module according to an exemplary embodiment of the present invention. 
     The input matrix processing module  110  according to an exemplary embodiment of the present invention includes a plurality of multiplexers. According to an exemplary embodiment of the present invention, the input matrix processing module  110  that processes a 4×4 matrix includes 10 multiplexers. 
     The elements a, b, c, and d constituting the first row vector in the matrix A expressed by Equation 1 are input to a first multiplexer  200  and a second multiplexer  210 . The first multiplexer  200  receives the elements a, b, and c and outputs “babaca” (hereinafter referred to as “first output”), and the second multiplexer  210  receives the elements b, c, and d and outputs “ccdddb” (hereinafter referred to as “second output”). 
     The elements e, f, g, and h constituting the second row vector are input to a third multiplexer  220 , a fourth multiplexer  230 , a fifth multiplexer  240 , and a sixth multiplexer  250 . The third multiplexer  220  and the fourth multiplexer  230  receive the elements e, f, and g, and the fifth multiplexer  240  and the sixth multiplexer  250  receive the elements f, g, and h. 
     Accordingly, the third multiplexer  220  outputs “fefege” (hereinafter referred to as “third output”), and the fourth multiplexer  230  outputs “efefeg” (hereinafter referred to as “fourth output”). In addition, the fifth multiplexer  240  outputs “gghhhf” (hereinafter referred to as “fifth output”), and the sixth multiplexer  250  outputs “hhggfh” (hereinafter referred to as “sixth output”). 
     The elements i, j, k, and l constituting the third row vector are input to a seventh multiplexer  260  and an eighth multiplexer  270 . The seventh multiplexer  260  receives the elements i, j, and k and outputs “ijijik” (hereinafter referred to as “seventh output”), and the eighth multiplexer  270  receives the elements j, k, and l and outputs “llkkjl” (hereinafter referred to as “eighth output”). 
     The elements m, n, o, and p constituting the fourth row vector are input to a ninth multiplexer  280  and a tenth multiplexer  290 . The ninth multiplexer  280  receives the elements m, n, and o and outputs “mnmnmo” (hereinafter referred to as “ninth output”), and the tenth multiplexer  290  receives the elements n, o, and p and outputs “ppoonp” (hereinafter referred to as “tenth output”). 
     As described above, the input matrix processing module  110  that receives the matrix A having the constituent elements expressed by Equation 1 can output the first to tenth outputs to the first cofactor operation module  120  and the second cofactor operation module  130  through a plurality of multiplexers. 
     The first cofactor operation module  120  and the second cofactor operation module  130  calculate the cofactors constituting the adjoint matrix adj(A), that is, c 11  to c 44 , on the basis of the first to tenth outputs from the input matrix processing module  110 . The first cofactor operation module  120  calculates the cofactors c 11  to c 24 , and the second cofactor operation module  130  calculates the cofactors c 31  to c 44 . 
     The structures and functions of the first cofactor operation module  120  and the second cofactor operation module  130  that calculate the cofactors will be described with reference to  FIGS. 3 and 6 . 
       FIG. 3  is a diagram schematically illustrating the internal configuration of a first cofactor operation module according to an exemplary embodiment of the present invention. 
     The first cofactor operation module  120  according to an exemplary embodiment of the present invention receives the first to third outputs, the fifth output, and the seventh to tenth outputs from the input matrix processing module  110 , and calculates the cofactors c 11  to c 24  from the first to third outputs, the fifth output, and the seventh to tenth outputs. 
     For this purpose, the first cofactor operation module  120  according to an exemplary embodiment of the present invention includes a plurality of multiplexers  300 ,  310 ,  330 ,  340 ,  350 , and  360 , a subtractor  320 , and two common factor generation modules  370  and  380 . 
     The eleventh multiplexer  300  receives the seventh output (ijijik) and the tenth output (ppoonp) from the input matrix processing module  110  and multiplexes them. In the description of the present invention, assuming the output of the eleventh multiplexer  300  is an eleventh output, the eleventh output becomes (ip jp io jo in kp). 
     The twelfth multiplexer  310  receives the eighth output (llkkjl) and the ninth output (mnmnmo) from the input matrix processing module  110  and multiplexes them. In the description of the present invention, assuming the output of the twelfth multiplexer  310  is a twelfth output, the twelfth output becomes (lm ln km kn jm lo). 
     The first subtractor  320  subtracts the cofactors, which are transmitted from the twelfth multiplexer  310 , from the cofactors, which are transmitted from the eleventh multiplexer  300 . Accordingly, assuming the output of the first subtractor  320  is a thirteenth output, the thirteenth output becomes (ip−lm jp−ln io−km jo−kn in−jm kp−lo). The first subtractor  320  transmits the thirteenth output to the thirteenth multiplexer  330  to the sixteenth multiplexer  360 . 
     The thirteenth multiplexer  330  receives the thirteenth output (ip−lm jp−ln io−km jo−kn in−jm kp−lo) from the first subtractor  320  and the fifth output (gghhhf), multiplexes them, and transmits the multiplexing result (fourteenth output) to the first common factor generation module  370 . 
     The fourteenth multiplexer  340  receives the thirteenth output (ip−lm jp−ln io−km jo−kn in−jm kp−lo) from the first subtractor  320  and the third output (fefege), multiplexes them, and transmits the multiplexing result (fifteenth output) to the first common factor generation module  370 . 
     The fifteenth multiplexer  350  receives the thirteenth output (ip−lm jp−ln io−km jo−kn in−jm kp−lo) from the first subtractor  320  and the second output (ccdddb), multiplexes them, and transmits the multiplexing result (sixteenth output) to the second common factor generation module  380 . 
     The sixteenth multiplexer  360  receives the thirteenth output (ip−lm jp−ln io−km jo−kn in−jm kp−lo) from the first subtractor  320  and the first output (babaca), multiplexes them, and transmits the multiplexing result (seventeenth output) to the second common factor generation module  380 . 
     The first common factor generation module  370  sums the fourteenth output and the fifteenth output to calculate common factors, and according to the calculation result, calculates the cofactors c 11  to c 14 . 
     The second common factor generation module  380  sums the sixteenth output and the seventeenth output to calculate common factors, and according to the calculation result, calculates the cofactors c 21  to c 24 . 
     The internal structures of the first common factor generation module  370  and the second common factor generation module  380  are as shown in  FIG. 4 . 
       FIG. 4  is a diagram schematically illustrating the internal configuration of a common factor generation module according to an exemplary embodiment of the present invention. 
     A common factor generation module includes a first sign conversion module  410 , a second sign conversion module  420 , an output line selection module  430 , and four output lines  440  to  470 . 
     The first sign conversion module  410  and the second sign conversion module  420  perform sign conversion by inverting the input factors and transmit the inverted factors to the output line selection module  430 , or transmit the factors to the output line selection module  430  as they are, without performing sign conversion. 
     The output line selection module  430  selects output lines to which the factors from the first sign conversion module  410  and the second sign conversion module  420  are output. According to an output line is selected by the output line selection module  430 , the outputs required for calculating the cofactors c 11  to c 14  can be selected. 
     The first output line  440  to the fourth output line  470  sum one or more outputs to be transmitted according to the selection by the output line selection module  430 , and transmit the summation result to the square calculation module  140 . 
       FIG. 5  shows diagrams illustrating the internal configurations of first common factor generation modules for calculating cofactors c 11  to c 14  adapted in various ways based on kinds of input factors according to an exemplary embodiment of the present invention. 
     In (A) of  FIG. 5 , the element f(kp−lo) is input to the first sign conversion module  410 , and the element e(kp−lo) is input to the second sign conversion module  420 . The first sign conversion module  410  and the second sign conversion module  420  do not perform sign conversion, and according to the selection by the output line selection module  430 , f(kp−lo) is output to the first output line  440  and e(kp−lo) is output to the second output line  450 . 
     In (B) of  FIG. 5 , the element h(in−jm) is input to the first sign conversion module  410 , and the element g(in−jm) is input to the second sign conversion module  420 . The first sign conversion module  410  and the second sign conversion module  420  do not perform sign conversion, and according to the selection by the output line selection module  430 , h(in−jm) is output to the third output line  460  and g(in−jm) is output to the fourth output line  470 . 
     In (C) of  FIG. 5 , the element h(jo−kn) is input to the first sign conversion module  410 , and the element e(jo−kn) is input to the second sign conversion module  420 . The first sign conversion module  410  and the second sign conversion module  420  do not perform sign conversion, and according to the selection by the output line selection module  430 , h(jo−kn) is output to the first output line  440  and e(jo−kn) is output to the fourth output line  470 . 
     In (D) of  FIG. 5 , the element h(jo−km) is input to the first sign conversion module  410 , and the element f(io−km) is input to the second sign conversion module  420 . In this case, while the first sign conversion module  410  does not perform sign conversion, the second sign conversion module  420  does perform sign conversion. Accordingly, h(io−km) is output to the second output line  450 , and −f(io−km) is output to the fourth output line  470 . 
     With the internal configuration of the first common factor generation module  370 , in (E) of  FIG. 5 , −g(jp−ln) is output to the first output line  440 , and e(jp−ln) is output to the third output line  460 . In addition, in (F) of  FIG. 5 , −g(ip−lm) is output to the second output line  450 , and −f(ip−lm) is output to the third output line  460 . 
     With the output results, each output line has the output result expressed by Equation 9.
 
First Output Line= f ( kp−lo )+ h ( jo−kn )− g ( jp−ln )
 
Second Output Line= e ( kp−lo )+ h ( io−km )− g ( ip−lm )
 
Third Output Line= h ( in−jm )+ e ( jp−ln )− f ( ip−lm )
 
Fourth Output Line= g ( in−jm )+ e ( jo−kn )− f ( io−km )  (Equation 9)
 
     Comparing the output results of the output lines expressed by Equation 9 with the cofactors of the adjoint matrix adj(A) calculated by Equation 8, the following can be seen: the output result of the first output line  440  is identical to the cofactor c 11 ; the output result of the second output line  450  is identical to the cofactor c 12 ; the output result of the third output line  460  is identical to the cofactor c 13 ; and the output result of the fourth output line  470  is identical to the cofactor c 14 . 
     That is, according to an exemplary embodiment of the present invention, the first common factor generation module  370  outputs the cofactors c 11  to c 14  through the output lines. 
     It can be seen that the second common factor generation module  380  having the same configuration as that of the first common factor generation module  370  outputs c 21  to c 24  through the output lines. 
     The outputs of the first common factor generation module  370  and the outputs of the second common factor generation module  380  become the outputs of the first cofactor operation module  120 . Accordingly, it can be seen that the outputs of the first cofactor operation module  120  are the cofactors c 11  to c 24 . 
       FIG. 6  is a diagram schematically illustrating the internal configuration of a second cofactor operation module according to an exemplary embodiment of the present invention. 
     The second cofactor operation module  130  according to an exemplary embodiment of the present invention receives the first and second outputs, the fourth output, and the sixth to tenth outputs from the input matrix processing module  110 , and calculates the cofactors c 31  to c 44  from the first and second outputs, the fourth output, and the sixth to tenth outputs. 
     For this purpose, the second cofactor operation module  130  according to an exemplary embodiment of the present invention includes a plurality of multiplexers  500 ,  510 ,  530 ,  540 ,  560 ,  560 ,  580 , and  590 , two subtractors  520  and  550 , and two common factor generation modules  600  and  610 . 
     The seventeenth multiplexer  500  receives the fourth output (efefeg) and the tenth output (ppoonp) from input matrix processing module  110 , and multiplexes them to generate an eighteenth output (ep fp eo fo en gp). 
     The eighteenth multiplexer  510  receives the sixth output (hhggfh) and the tenth output (mnmnmo) from the input matrix processing module  110 , and multiplexes them to generate a nineteenth output (hm hn gm gn fm ho). 
     The second subtractor  520  subtracts the cofactors that are transmitted from the eighteenth multiplexer  510  from the cofactors that are transmitted from the seventeenth multiplexer  500 . The output of the second subtractor  520 , that is, a twentieth output (ep−hm fp−hn eo−gm fo−gn en−fm gp−ho), is transmitted to the twenty-first multiplexer  560  and the twenty-second multiplexer  570 . 
     The nineteenth multiplexer  530  receives the fourth output (efefeg) and the eighth output (llkkjl) from the input matrix processing module  110 , and multiplexes them to generate a twenty-first output (el fl ek fk ej gl). 
     The twentieth multiplexer  540  receives the sixth output (hhggfh) and the seventh output (ijijik) from the input matrix processing module  110 , and multiplexes them to generate a twenty-second output (hi hj gi gj fi hk). 
     The third subtractor  550  subtracts the cofactors that are transmitted from the twentieth multiplexer  540  from the cofactors that are transmitted from the nineteenth multiplexer  530 . The output of the third subtractor  550 , that is, a twenty-third output (el−hi fl−hj ek−gi fk−gj ej−fi gl−hk), is transmitted to the twenty-third multiplexer  580  and the twenty-fourth multiplexer  590 . 
     The twenty-first multiplexer  560  receives the twentieth output (ep−hm fp−hn eo−gm fo−gn en−fm gp−ho) from the second subtractor  520  and the second output (ccdddb), multiplexes them, and transmits the multiplexing result (twenty-fourth output) to the third common factor generation module  600 . 
     The twenty-second multiplexer  570  receives the twentieth output (ep−hm fp−hn eo−gm fo−gn en−fm gp−ho) from the second subtractor  520  and the first output (babaca), multiplexes them, and transmits the multiplexing result (twenty-fifth output) to the third common factor generation module  600 . 
     The twenty-third multiplexer  580  receives the twenty-third output (el−hi fl−hj ek−gi fk−gj ej−fi gl−hk) from the third subtractor  520  and the second output (ccdddb), multiplexes them, and transmits the multiplexing result (twenty-sixth output) to the fourth common factor generation module  610 . 
     The twenty-fourth multiplexer  590  receives the twenty-third output (el−hi fl−hj ek−gi fk−gj ej−fi gl−hk) from the third subtractor  520  and the first output (babaca), multiplexes them, and transmits the multiplexing result (twenty-seventh output) to the fourth common factor generation module  610 . 
     The third common factor generation module  600  sums the twenty-fourth output and the twenty-fifth output to calculate common factors, and according to the calculation result, calculates the cofactors c 31  to c 34 . 
     The fourth common factor generation module  610  sums the twenty-sixth output and the twenty-seventh output to calculate common factors, and according to the calculation result, calculates the cofactors c 41  to c 44 . 
     The third common factor generation module  600  and the fourth common factor generation module  610  have the same internal structure and function as the first common factor generation module  370  and the second common factor generation module  380  shown in  FIGS. 4 and 5A  to  5 F, and thus detailed descriptions thereof will be omitted. 
     It can be seen that the cofactors c 31  to c 44  are output from the second cofactor operation module  130 . 
     The cofactors c 11  to c 24  output through the output lines of the first cofactor operation module  120  shown in  FIG. 3  are input to the square calculation module  140 . 
     The cofactors c 31  to c 44  output through the output lines of the second cofactor operation module  130  shown in  FIG. 6  are also input to the square calculation module  140 . 
     The square calculation module  140  squares the factors transmitted through the output lines, and transmits the squared factors to the summation module  150 . At this time, the square calculation module  140  independently processes the factors transmitted from the common factor generation modules  370 ,  380 ,  600 , and  610 , and transmits the processing results to the summation module  150 . 
     That is, the square calculation module  140  receives and squares the cofactors c 11  to c 14  transmitted from the first common factor generation module  370  of the first cofactor operation module  120 , and transmits the squared factors c 11   2  to c 14   2  to the summation module  150  as a twenty-eighth output. In addition, the square calculation module  140  receives and squares the cofactors c 21  to c 24  transmitted from the second common factor generation module  380 , and transmits the squared factors c 21   2  to c 24   2  to the summation module  150  as a twenty-ninth output. 
     Also, the square calculation module  140  receives and squares the cofactors c 31  to c 34  transmitted from the third common factor generation module  600  of the second cofactor operation module  130 , and transmits the squared factors c 31   2  to c 34   2  to the summation module  150  as a thirtieth output. In addition, the square calculation module  140  receives and squares the cofactors c 41  to c 44  transmitted from the fourth common factor generation module  610 , and transmits the squared factors c 41   2  to c 44   2  to the summation module  150  as a thirty-first output. 
     The summation module  150  selects four factors among the square results (c 11   2  to c 14   2 , c 21   2  to c 24   2 , c 31   2  to c 34   2 , and c 41   2  to c 44   2 ) from the square calculation module  140 , sums the selected factors, and transmits the summation result to the norm comparison module  160 . At this time, the summation module  150  is configured to select the factors at the same position of the twenty-eighth output to the thirty-first output. 
     For example, the summation module  150  selects the factors c 11 , c 21  c 31   2 , and c 41   2  at the first position of the twenty-eighth output to the thirty-first output from the square calculation module  140 , and transmits the summation result (c 11   2 +c 21   2 +c 31   2 +c 41   2 ) of the selected factors to the norm comparison module  160  as a thirty-second output. 
     Also, the summation module  150  selects the factors c 12   2 , c 22   2 , c 32   2 , and c 42   2  at the second position of the twenty-eighth output to the thirty-first output, and the summation result (c 12   2 +c 22   2 +c 32   2 +c 42   2 ) of the selected factors to the norm comparison module  160  as a thirty-third output. 
     Similarly, the summation module  150  transmits a thirty-fourth output (c 13   2 +c 23   2 +c 33   2 +c 43   2 ) and a thirty-fifth output (c 14   2 +c 24   2 +c 34   2 +c 44   2 ) to the norm comparison module  160 . 
     It can be seen that the thirty-second output to the thirty-fifth output transmitted to the norm comparison module  160  through the summation module  150  are correspondingly identical to the norms of the row vectors in the adjoint matrix adj(A) expressed by Equations 4 to 7. That is, the thirty-second output is identical to the norm of the first row vector in the adjoint matrix adj(A), and the thirty-third output is identical to the norm of the second row vector therein. In addition, the thirty-fourth output is identical to the norm of the third row vector in the adjoint matrix adj(A), and the twenty-fifth output is identical to the norm of the fourth row vector therein. 
       FIG. 7  is a diagram schematically illustrating the internal configuration of a norm comparison module according to an exemplary embodiment of the present invention. 
     The norm comparison module  160  includes a plurality of comparators. In the row-vector norm comparison apparatus for an inverse matrix of a 4×4 matrix according to an exemplary embodiment of the present invention, the norm comparison module  160  includes, for example, five comparators  710  to  750 . 
     The first comparator  710  receives the thirty-second output and the thirty-third output from the summation module  150 , and compares the norms of the first and second row vectors. 
     The second comparator  720  receives the thirty-fourth output and the thirty-fifth output from the summation module  150 , and compares the norms of the third and fourth row vectors. 
     Although a case in which the first comparator  710  receives and compares the thirty-second output and the thirty-third output, and the second comparator  720  receives and compares the thirty-fourth output and the thirty-fifth output has been described, the first comparator  710  and the second comparator  720  may be adapted such that any of the thirty-second output to the thirty-fifth output are input to the first comparator  710  and the second comparator  720 . 
     The third comparator  730  selects a row vector having the biggest norm by comparing a row vector having a bigger norm according to the comparison result of the first comparator  710  with a row vector having a bigger norm according to the comparison result of the second comparator  710 . 
     The fourth comparator  740  selects a row vector having the smallest norm by comparing a row vector having a smaller norm according to the comparison result of the first comparator  710  with a row vector having a smaller norm according to the comparison result of the second comparator  710 . 
     The fifth comparator  750  selects row vectors having the second biggest norm and the third biggest norm by comparing a row vector having a smaller norm according to the comparison result of the third comparator  730  with a row vector having a bigger norm according to the comparison result of the fourth comparator  740 . 
     With the results of the norm comparison module  160 , it is possible to confirm the row-vector norm comparison result for the inverse matrix of the 4×4 matrix. 
     The embodiment of the present invention described above is not be implemented by only the method and apparatus, but it may be implemented by a program for executing the functions corresponding to the configuration of the exemplary embodiment of the present invention or a recording medium having the program recorded thereon. These implementations can be realized by the ordinarily skilled person in the art from the description of the above-described exemplary embodiment. 
     While this invention has been described in connection with what is presently considered to be practical exemplary embodiments, it is to be understood that the invention is not limited to the disclosed embodiments, but, on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.