an iterative method for the skew symmetric solution and the optimal approximate solution of the matrix equation axb c in this paper an iterative method is constructed to solve the linear matrix equation axb c over skew symmetric matrix x by the iterative method the solvability of the equation axb c over skew symmetric matrix can be determined automatically when the equation axb c is consistent over skew symmetric matrix x for any skew symmetric initial iterative matrix x 1 the skew symmetric solution can be obtained within finite iterative steps in the absence of roundoff errors the unique least norm skew symmetric iterative solution of axb c can be derived when an appropriate initial iterative matrix is chosen a sufficient and necessary condition for whether the equation axb c is inconsistent is given furthermore the optimal approximate solution of axb c for a given matrix x 0 can be derived by finding the least norm skew symmetric solution of a new corresponding matrix equation email protected email protected finally several numerical examples are given to illustrate that our iterative method is effective

gradient based iterative algorithm for solving coupled matrix equations abstract this paper is concerned with iterative methods for solving a class of coupled matrix equations including the well known coupled markovian jump lyapunov matrix equations as special cases the proposed method is developed from an optimization point of view and contains the well known jacobi iteration gauss seidel iteration and some recently reported iterative algorithms by using the hierarchical identification principle as special cases we have provided analytically the necessary and sufficient condition for the convergence of the proposed iterative algorithm simultaneously the optimal step size such that the convergence rate of the algorithm is maximized is also established in explicit form the proposed approach requires less computation and is numerically reliable as only matrix manipulation is required some other existing results require either matrix inversion or special matrix products numerical examples show the effectiveness of the proposed algorithm

on preconditioned iterative methods for burgers equations we study the newton method and a fixed point method for solving the system of nonlinear equations arising from the sinc galerkin discretization of the burgers equations in each step of the newton method or the fixed point method a structured subsystem of linear equations is obtained and needs to be solved numerically in this paper preconditioning techniques are applied to solve such linear subsystems the bounds for eigenvalues of the preconditioned matrices are derived and numerical examples are given to illustrate the effectiveness of the proposed methods we also find that a combination of the newton fixed point iteration with the preconditioned gmres method is quite efficient for the sinc galerkin discretization of the burgers equations

geometric mapping properties of semipositive matrices abstract semipositive matrices map a positive vector to a positive vector and as such they are a very broad generalization of the irreducible nonnegative matrices nevertheless the ensuing geometric mapping properties of semipositive matrices result in several parallels to the theory of cone preserving and cone mapping matrices it is shown that for a semipositive matrix a there exist a proper polyhedral cone k 1 of nonnegative vectors and a polyhedral cone k 2 of nonnegative vectors such that a k 1 k 2 the set of all nonnegative vectors mapped by a to the nonnegative orthant is a proper polyhedral cone as a consequence a belongs to a proper polyhedral cone comprising semipositive matrices when the powers a k k 0 1 have a common semipositivity vector then a has a positive eigenvalue if a has a sole peripheral eigenvalue and the powers of a have a common semipositivity vector with a non vanishing term in the direction of the left eigenspace of then a leaves a proper cone invariant

a modified gradient based algorithm for solving sylvester equations abstract in this paper a modified gradient based algorithm for solving sylvester equations is presented different from the gradient based method introduced by ding and chen 7 and the relaxed gradient based algorithm proposed by niu et al 18 the information generated in the first half iterative step is fully exploited and used to construct the approximate solution theoretical analysis shows that the new method converges under certain assumptions numerical results are given to verify the efficiency of the new method

successive projection iterative method for solving matrix equation ax b in this paper we present a new iterative method successive projection iterative method to solve matrix equation ax b where a is a symmetric positive definite spd matrix based on this method an algorithm is proposed and proved to be convergent in addition analysis of the algorithm and numerical experiments are also given to show the efficiency of the method

square matrix embeddable space time block codes for complex signal constellations space time block codes for providing transmit diversity in wireless communication systems are considered based on the principles of linearity and unitarity a complete classification of linear codes is given in the case when the symbol constellations are complex and the code is based on a square matrix or restriction of such by deleting columns antennas maximal rate delay optimal codes are constructed within this category the maximal rates allowed by linearity and unitarity fall off exponentially with the number of transmit antennas

gradient based iterative solutions for general linear matrix equations in this paper we present a gradient based iterative algorithm for solving general linear matrix equations by extending the jacobi iteration and by applying the hierarchical identification principle convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices two numerical examples are provided to show that the proposed algorithm is effective

iterative solutions of the generalized sylvester matrix equations by using the hierarchical identification principle abstract in this paper by extending the well known jacobi and gauss seidel iterations for ax b we study iterative solutions of matrix equations axb f and generalized sylvester matrix equations axb cxd f including the sylvester equation ax xb f as a special case and present a gradient based and a least squares based iterative algorithms for the solution it is proved that the iterative solution always converges to the exact solution for any initial values the basic idea is to regard the unknown matrix x to be solved as the parameters of a system to be identified and to obtain the iterative solutions by applying the hierarchical identification principle finally we test the algorithms and show their effectiveness using a numerical example

the general coupled matrix equations over generalized bisymmetric matrices abstract in the present paper by extending the idea of conjugate gradient cg method we construct an iterative method to solve the general coupled matrix equations j 1 p a ij x j b ij m i i 1 2 p including the generalized coupled lyapunov and sylvester matrix equations as special cases over generalized bisymmetric matrix group x 1 x 2 x p by using the iterative method the solvability of the general coupled matrix equations over generalized bisymmetric matrix group can be determined in the absence of roundoff errors when the general coupled matrix equations are consistent over generalized bisymmetric matrices a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors the least frobenius norm generalized bisymmetric solution group of the general coupled matrix equations can be derived when an appropriate initial iterative matrix group is chosen in addition the optimal approximation generalized bisymmetric solution group to a given matrix group x 1 x 2 x p in frobenius norm can be obtained by finding the least frobenius norm generalized bisymmetric solution group of new general coupled matrix equations the numerical results indicate that the iterative method works quite well in practice

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

principal pivot transforms properties and applications the principal pivot transform ppt is a transformation of the matrix of a linear system tantamount to exchanging unknowns with the corresponding entries of the right hand side of the system the notion of the ppt is encountered in mathematical programming statistics and numerical analysis among other areas the purpose of this paper is to draw attention to the main properties and uses of ppts make some new observations and motivate further applications of ppts in matrix theory special consideration is given to ppts of matrices whose principal minors are positive 2000 elsevier science inc all rights reserved

iterative solutions to matrix equations of the form aixbi fi this paper is concerned with the numerical solutions to the linear matrix equations a 1xb 1 f 1 and a 2xb 2 f 2 two iterative algorithms are presented to obtain the solutions for any initial value it is proved that the iterative solutions obtained by the proposed algorithms converge to their true values finally simulation examples are given to verify the proposed convergence theorems

a representation of the general common solution to the matrix equations a1xb1 c1 and a2xb2 c2 with applications we give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations a1xb1 c1 and a2xb2 c2 and derive a new representation of the general common solution to these two equations we apply this result to determine new necessary and sufficient conditions for the existence of an hermitian solution and a representation of the general hermitian solution to the matrix equation axb c

the mgpbicg method for solving the generalized coupled sylvester conjugate matrix equations in this paper we extend the generalized product type bi conjugate gradient gpbicg method for solving the generalized sylvester conjugate matrix equations a 1 x b 1 c 1 y d 1 s 1 a 2 x b 2 c 2 y d 2 s 2 by the real representation of the complex matrix and the properties of kronecker product and vectorization operator some numerical experiments demonstrate that the introduced iteration approach is efficient

ranks and the least norm of the general solution to a system of quaternion matrix equations we in this paper first establish a new expression of the general solution to the consistent system of linear quaternion matrix equations a1x1 c1 a2x2 c2 a3x1b1 a4x2b2 c3 which was investigated recently q w wang h x chang c y lin p skew symmetric common solutions to a pair of quaternion matrix equations appl math comput 195 2008 721 732 then derive the maximal and minimal ranks and the least norm of the general solution to the system mentioned above some previous known results can be viewed as special cases of the results of this paper

an iterative algorithm for the reflexive solutions of the generalized coupled sylvester matrix equations and its optimal approximation abstract the generalized coupled sylvester matrix equations ay zb cy zd e f with unknown matrices y z are encountered in many systems and control applications also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils in this work we construct an iterative algorithm to solve the generalized coupled sylvester matrix equations over reflexive matrices y z and when the matrix equations are consistent for any initial matrix pair y 0 z 0 a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors and the least frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair also we obtain the optimal approximation reflexive solution pair to a given matrix pair y z in the reflexive solution pair set of the generalized coupled sylvester matrix equations ay zb cy zd e f moreover several numerical examples are given to show the efficiency of the presented iterative algorithm

iterative algorithms for the minimum norm solution and the least squares solution of the linear matrix equations abstract in this paper two iterative algorithms are proposed to solve the linear matrix equations a 1 xb 1 c 1 x t d 1 m 1 a 2 xb 2 c 2 x t d 2 m 2 when the matrix equations are consistent by the first algorithm a solution x can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore the minimum norm solution can be got by choosing a special kind of initial matrix additionally the unique optimal approximation solution to a given matrix x 0 can be derived by finding the minimum norm solution of a new matrix equations a 1 x b 1 c 1 x t d 1 m 1 a 2 x b 2 c 2 x t d 2 m 2 when the matrix equations are inconsistent we present the second algorithm to find the least squares solution with the minimum norm finally two numerical examples are tested by matlab the results show that these iterative algorithms are efficient

results concerning interval linear systems with multiple right hand sides and the interval matrix equation ax b this note tries to study different solution sets of the interval linear matrix equation ax b where a is a known square interval matrix of dimension mxm b is a rectangular interval matrix of dimension mxn while the unknown matrix x is also of dimension mxn firstly we show that shary s results for interval linear systems with a single right hand side vector cannot be simply generalized to the case of interval linear systems of the form ax b secondly we give some analytical characterizations of the ae solution sets of this interval matrix equation we use a linear programming method in order to find the interval hull matrix on the other hand we propose the use of an interval gaussian elimination to find an enclosure for the united solution set of this matrix equation since the lu decomposition of a is needed only once numerical examples have also been given

solutions to a family of matrix equations by using the kronecker matrix polynomials closed form solutions to a family of generalized sylvester matrix equation in form of i 0 fa ixf email protected i 0 jb kyf email protected j 0 fe jrf j are given by using the so called kronecker matrix polynomials it is found that the structure of the solutions is independent of the orders f j and f this type of uniform closed form solutions includes our early results as special cases the results provide great convenience to the computation and analysis of the solutions to this class of equations and can perform important functions in many analysis and design problems in linear systems

some results on matrix symmetries and a pattern recognition application this correspondence deals with various matrix symmetries frequently encountered in practice it is shown that the class of doubly symmetric matrices is not closed under multiplication whereas centrosymmetric matrices form an abelian group under addition and a non abelian group under multiplication results similar to centrosymmetric matrices are presented for the special case of centro hermitian matrices a brief discussion of a pattern recognition feature selection problem is included to demonstrate the manner in which these results may be utilized in practice for developing computationally efficient algorithms

the accelerated gradient based iterative algorithm for solving a class of generalized sylvester transpose matrix equation in this paper we present an accelerated gradient based algorithm by minimizing certain criterion quadratic function for solving the generalized sylvester transpose matrix equation a x b c x t d f the idea is from ding and chen 2005 niu et al 2011 wang et al 2012 in which some efficient algorithms were developed for solving the sylvester matrix equation and the lyapunov matrix equation on the basis of the information generated in the previous half step we further introduce a relaxation factor to obtain the solution of the generalized sylvester transpose matrix equation we show that the iterative solution converges to the exact solution for any initial value provided that some appropriate assumptions finally some numerical examples are given to illustrate that the introduced iterative algorithm is efficient

a new iteration method for the matrix equation ax b an iteration method for the matrix equation ax b is constructed by this iteration method the least norm solution for the matrix equation can be obtained when the matrix equation is consistent and the least norm least squares solutions can be obtained when the matrix equation is not consistent the related optimal approximation solution is also obtained by this iteration method a preconditioned method for improving the iteration rate is put forward finally some numerical examples are given

the common solution to six quaternion matrix equations with applications abstract we in this paper establish necessary and sufficient conditions for the existence of and an explicit expression for a common solution to six classical linear quaternion matrix equations a 1 x c 1 xb 1 c 2 a 2 x c 3 xb 2 c 4 a 3 xb 3 c 5 and a 4 xb 4 c 6 as applications symmetric persymmetric and centrosymmetric solutions to certain systems of quaternion matrix equations are presented furthermore we give an example to illustrate the result of this paper some known results can be regarded as the special cases of this paper

gradient based and least squares based iterative algorithms for matrix equations axb cxtd f abstract this paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation axb cx t d f the basic idea is to decompose the matrix equation system under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving ax b and axb f the analysis shows that when the matrix equation has a unique solution under the sense of least squares the iterative solution converges to the exact solution for any initial values a numerical example verifies the proposed theorems

gradient based iterative algorithms for solving a class of matrix equations in this note we apply a hierarchical identification principle to study solving the sylvester and lyapunov matrix equations in our approach we regard the unknown matrix to be solved as system parameters to be identified and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions we prove that the iterative solution consistently converges to the true solution for any initial value and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor or step size appropriately furthermore the iterative method is extended to solve general linear matrix equations the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

least squares solutions to ax b for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation abstract a matrix a r n n is called a bisymmetric matrix if its elements a i j satisfy the properties a i j a j i and a i j a n j 1 n i 1 for 1 i j n this paper considers least squares solutions to the matrix equation ax b for a under a central principal submatrix constraint and the optimal approximation a central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix we first discuss the specified structure of bisymmetric matrices and their central principal submatrices then we give some necessary and sufficient conditions for the solvability of the least squares problem and derive the general representation of the solutions moreover we also obtain the expression of the solution to the corresponding optimal approximation problem

on positive definite and skew hermitian splitting iteration methods for continuous sylvester equation ax xb c in this paper we present a positive definite and skew hermitian splitting pss iteration method for continuous sylvester equations ax xb c with positive definite semi definite matrices the theoretical analysis shows that the pss iteration method will converge unconditionally and the optimal parameter of the new method is presented moreover to reduce the computing cost an inexact variant of the pss iteration method ipss and the analysis of its convergence property in detail have been established numerical results show that this new method and its inexact invariant are efficient and robust solvers for this class of continuous sylvester equations

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

on the generalized reflexive and anti reflexive solutions to a system of matrix equations let p and q be two generalized reflection matrices i e p ph p2 i and q qh q2 i an n n matrix a is said to be generalized reflexive generalized anti reflexive with respect to the matrix pair p q if a paq a paq it is obvious that any n m matrix is also a generalized reflexive with respect to the matrix pair in im by extending the conjugate gradient least square cgls approach the present paper treats two iterative algorithms to solve the system of matrix equations f1 x a1 f2 x a2 fm x am including the sylvester and lyapunov matrix equations as special cases over the generalized reflexive and anti reflexive matrices where f1 f2 fm are the linear operators from cn n onto cri si and ai cri si for i 1 2 m when this system is consistent over the generalized reflexive generalized anti reflexive matrix it is proved that the first second iterative algorithm converges to a generalized reflexive generalized anti reflexive solution for any initial generalized reflexive generalized anti reflexive matrix also the first second iterative algorithm can obtain the the least frobenius norm generalized reflexive generalized anti reflexive solution for special initial generalized reflexive generalized anti reflexive matrix furthermore the optimal approximation generalized reflexive generalized anti reflexive solution to a given generalized reflexive generalized anti reflexive matrix can be derived by finding the least frobenius norm generalized reflexive generalized anti reflexive solution of a new system of matrix equations finally we test the proposed iterative algorithms and show their effectiveness using numerical examples

the block least squares method for solving nonsymmetric linear systems with multiple right hand sides abstract in this paper we present the block least squares method for solving nonsymmetric linear systems with multiple right hand sides this method is based on the block bidiagonalization we first derive two algorithms by using two different convergence criteria the first one is based on independently minimizing the 2 norm of each column of the residual matrix and the second approach is based on minimizing the frobenius norm of residual matrix we then give some properties of these new algorithms finally some numerical experiments on test matrices from harwell boeing collection are presented to show the efficiency of the new method

global least squares method gl lsqr for solving general linear systems with several right hand sides abstract in this paper we propose a new method for solving general linear systems with several right hand sides this method is based on global least squares method and reduces the original matrix to the lower bidiagonal form we derive a simple recurrence formula for generating the sequence of approximate solutions x k some theoretical properties of the new method are discussed and we also show that how this method can be implemented for the sylvester equation finally some numerical experiments on test matrices are presented to show the efficiency of the new method

geometric conditions for the existence of positive eigenvalues of matrices finite dimensional theorems of perron frobenius type are proved for a e e nn and a nonnegative integer k we let w t a be the cone generated by ai ak l in en we show that a satisfies the perron schaefer condition if and only if the closure wk a of w t a is a pointed cone this theorem is closely related to several known results if k po a the index of the eigenvalue 0 in spec a we prove that a has a positive eigenvalue if and only if w t a is a pointed nonzero cone or equivalently w t a is not a real subspace of en our proofs are elementary and based on a method of birkhoff s we discuss the relation of this method to pringsheim s theorem