a new representation for at s 2 3 a new representation of a t s 2 3 of a having prescribed range space t and null space s is derived using this representation the well known generalized inverses such as the moore penrose inverse the group inverse the drazin inverse the bott duffin inverse and the generalized bott duffin inverse are computed three numerical examples are worked out to demonstrate the computation of the moore penrose inverse the drazin inverse and the 2 3 inverse to show the efficacy of our approach

innovation based on gaussian elimination to compute generalized inverse at s 2 abstract in this paper we execute elementary row and column operations on the partitioned matrix g a g g g 0 into i s 0 0 0 0 0 a t s 2 to compute generalized inverse a t s 2 of a given complex matrix a where g is a matrix such that r g t and n g s the total number of multiplications and divisions operations is t m n s 2 m n 2 4 m s 1 2 n s m s n s m n s and the upper bound of t m n s is less than 6 m n 2 3 2 n 3 1 2 n 2 when n m a numerical example is shown to illustrate that this method is correct

on integral representation of the generalized inverse at s 2 we present a general integral representation for the generalized inverse a t s 2 which extends earlier result on the moore penrose inverse weighted moore penrose inverse and drazin inverse

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

recurrent neural networks for computing weighted moore penrose inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 three recurrent neural networks are presented for computing the weighted moore penrose inverse of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of weighted moore penrose inverse under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a weighted moore penrose inverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a weighted moore penrose inverse and a lagrangian matrix with constant connections

a solution of gaussian optimizer conjecture for quantum channels the long standing conjectures of the optimality of gaussian inputs and additivity are solved for a broad class of gauge covariant or contravariant bosonic gaussian channels which includes in particular thermal additive classical noise and amplifier channels restricting to the class of states with finite second moments we show that the vacuum is the input state which minimizes the entropy at the output of such channels this allows us to show also that the classical capacity of these channels under the input energy constraint is additive and is achieved by gaussian encodings

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

the reverse order law for 1 3 4 inverse of the product of two matrices the relationship between 1 3 4 inverses of ab and the product of 1 3 4 inverses of a and b have been studied in this paper the necessary and sufficient conditions for b 1 3 4 a 1 3 4 ab 1 3 4 b 1 3 4 a 1 3 4 ab 1 3 4 and b 1 3 4 a 1 3 4 ab 1 3 4 are given

a finite algorithm for computing the weighted moore penrose inverse a mn this paper presents a finite algorithm for computing the weighted moore penrose inverse of a rectangular matrix this algorithm may be viewed as an extension of decell s algorithm

theory and numerics of three dimensional beams with elastoplastic material behaviour a theory of space curved beams with arbitrary cross sections and an associated finite element formulation is presented within the present beam theory the reference point the centroid the centre of shear and the loading point are arbitrary points of the cross section the beam strains are based on a kinematic assumption where torsion warping deformation is included each node of the derived finite element possesses seven degrees of freedom the update of the rotational parameters at the finite element nodes is achieved in an additive way applying the isoparametric concept the kinematic quantities are approximated using lagrangian interpolation functions since the reference curve lies arbitrarily with respect to the centroid the developed element can be used to discretize eccentric stiffener of shells due to the implemented constitutive equations for elastoplastic material behaviour the element can be used to evaluate the load carrying capacity of beam structures copyright 2000 john wiley sons ltd

comments on some recent results concerning 2 3 and 2 4 generalized inverses abstract we give a comment on some recent results concerning the representations of generalized 2 3 and 2 4 inverses shorter proofs of some previous results are presented

the representation of generalized inverse at s 2 3 and its applications this paper presents the explicit expression for matrix right symmetry factor with prescribed rang and null space moreover the explicit expression for generalized inverse a t s 2 3 which is a 2 3 inverse of a having the prescribed rang t and null space s is derived as an application two numerical examples are given

successive matrix squaring algorithm for parallel computing the weighted generalized inverse amn we derive a successive matrix squaring sms algorithm to approximate the weighted generalized inverse which can be expressed in the form of successive squaring of a composite matrix t given an m by n matrix a with m n we show that the weighted generalized inverse of a can be computed in parallel time ranging from o log n to o log 2n provided that there are enough processors to support matrix multiplication in time o log n

methods for estimating capacities and rates of gaussian quantum channels optimization methods aimed at estimating the capacities of a general gaussian channel are developed specifically evaluation of classical capacity as maximum of the holevo information is pursued over all possible gaussian encodings for the lossy bosonic channel but extension to other capacities and other gaussian channels seems feasible solutions for both memoryless and memory channels are presented it is first dealt with single use single mode channel where the capacity dependence on channel s parameters is analyzed providing the full classification of possible cases then it is dealt with multiple uses multimode channel where the capacity dependence on the multimode environment state is analyzed when both total environment energy and environment purity are fixed this allows a fair comparison among different environments thus understanding the role of memory intermode correlations and phenomenon like superadditivity of the capacity the developed methods are also used for deriving transmission rates with heterodyne and homodyne measurements at the channel output classical capacity and transmission rates are presented within a unique framework where the rates can be treated as logarithmic approximations of the capacity

the representation and computation of generalized inverse at s 2 this paper presents a novel representation for the generalized inverse a t s 2 based on this we give an algorithm to compute this generalized inverse as an application we use gauss jordan elimination to compute the weighted moore penrose inverse a m n and the drazin inverse a d

computing 2 4 and 2 3 inverses by using the sherman morrison formula a finite recursive procedure for computing 2 4 generalized inverses and the analogous recursive procedure for computing 2 3 generalized inverses of a given complex matrix are presented the starting points of both introduced methods are general representations of these classes of generalized inverses these representations are formed using certain matrix products which include the moore penrose inverse or the usual inverse of a symmetric matrix product and the sherman morrison formula for the inverse of a symmetric rank one matrix modification the computational complexity of the methods is analyzed defined algorithms are tested on randomly generated matrices as well as on test matrices from the matrix computation toolbox

matrix left symmetry factor and its applications in generalized inverses a 2 4 t s in this paper we propose some definitions about matrix symmetry factors and discuss the construction of matrix left symmetry factor in detail the explicit expression of the generalized inverses at s 2 4 which is a 2 4 inverses of a having the prescribed rang t and null space s is derived furthermore using the expression at s 2 4 we get the corresponding results for six important generalized inverses

full rank representations of 2 4 2 3 inverses and successive matrix squaring algorithm abstract we present the full rank representations of 2 4 and 2 3 inverses with given rank as well as with prescribed range and null space as particular cases of the full rank representation of outer inverses as a consequence two applications of the successive matrix squaring sms algorithm from p s stanimirovic d s cvetkovic ilic successive matrix squaring algorithm for computing outer inverses appl math comput 203 2008 19 29 are defined using the full rank representations of 2 4 and 2 3 inverses the first application is used to approximate 2 4 inverses the second application after appropriate modifications of the sms iterative procedure computes 2 3 inverses of a given matrix presented numerical examples clarify the purpose of the introduced methods

the generalized bott duffin inverse and its applications abstract the generalized bott duffin inverse a l of a with respect to a subspace l is defined and by means of it a unified treatment of a class of linear systems and extreme value problems with equality constraints is given