about the group inverse and moore penrose inverse of a product abstract let a be a matrix over a ring suppose that p q p and q are matrices over the ring such that p pa a aqq if a has group inverse a then paq has a group inverse iff aa qpa 1 aa is invertible an analogous result for moore penrose inverses is also given

the linear complementarity problem with exact order matrices dedicated to professor k g ramamurthy on the occasion of his 60th birthday a real n by n matrix a is called an np matrix of exact order k if the principal minors of a of order 1 through n k are negative positive and n k 1 through n are positive negative in this paper the properties of exact order 1 and 2 matrices are investigated using the linear complementarity problem lcpq a for each q rn a complete characterization of the class of exact order 1 based on the number of solutions to the lcpq a for each q rn is presented in the last season we consider the problem of computing a solution to the lcpq a when a is a matrix of exact order 1 or 2

a comment on some recent results concerning the reverse order law for 1 3 4 inverses abstract this paper has been motivated by the one of liu and yang d liu h yang the reverse order law for 1 3 4 inverse of the product of two matrices appl math comp 215 12 2010 4293 4303 in which the authors consider separately the cases when ab 1 3 4 b 1 3 4 a 1 3 4 and ab 1 3 4 b 1 3 4 a 1 3 4 where a c n m and b c m n here we prove that ab 1 3 4 b 1 3 4 a 1 3 4 is actually equivalent to ab 1 3 4 b 1 3 4 a 1 3 4 we show that ab 1 3 4 b 1 3 4 a 1 3 4 can only be possible if n m and in this case we present purely algebraic necessary and sufficient conditions for this inclusion to hold also we give some new characterizations of b 1 3 4 a 1 3 4 ab 1 3 4

on parallel sum of operators abstract the basic properties of the parallel sum were developed for nonnegative matrices in finite dimensional spaces however without suitable restrictions very little of the preceding theories will hold for bounded linear operators a and b acting in hilbert space in this paper generalization to non selfadjoint operators is considered and various properties of parallel sum a b a a b b are given meanwhile the common upper and lower bounds of positive operators by using the parallel sum are given and some relationships between the parallel sum of projection operators are obtained

the boolean pivot operation m matrices and reducible matrices abstract let a be a boolean n n matrix and let g n u be the corresponding diagraph where n 1 n is the set of vertices and u n n is the set of arcs of g for any r n combining the steps r r of warshall s algorithm for determining the reachability matrix of a yields the boolean pivot operation b r the matrix b r a is the so called r reachability matrix of a a ij 1 if and only if aij 1 or there is a connection between i and j via vertices belonging to r we also have b r b s b s b r b r s for any r s n the boolean pivot operation is closely related to the principal pivotal operation for real matrices so we obtain the boolean analogues to the formula for inverting a real block matrix and for the sherman morrison woodbury formulae for updating the inverse of a real matrix using the close connection between nonsingular m matrices and the corresponding boolean matrices we obtain a flexible algorithm for excluding and including vertices one at a time in g while retaining the original connections between the vertices of the current diagraph we derive some criteria for irreducibility of boolean block matrices and of partitioned z matrices and give a condensed form of warshall s algorithm for testing whether a boolean matrix is irreducible moreover we correct two recent results by r l smith on irreducibility of real block matrices

mixed type reverse order laws for 1 3 4 generalized inverses over hilbert spaces abstract the reverse order laws for 1 3 4 generalized inverses of a product of two operators have been studied by wang et al j wang h zhang g ji a generalized reverse order law for the products of two operators journal of shaanxi normal university 38 4 2010 13 17 in this paper using a block operator matrix technique we study mixed type reverse order laws for 1 3 4 generalized inverses over hilbert spaces

reverse order law for reflexive generalized inverses of products of matrices abstract the relationships between generalized inverses of the product of two matrices a b and the product of generalized inverses of a b have been studied in the literature by applying the product singular value decomposition p svd in this paper we derive equivalent conditions for b 1 2 a 1 2 ab 1 2 ab 1 2 b 1 2 a 1 2 b 1 2 a 1 2 ab 1 2 and equivalent conditions for ab 1 b 1 a 1

a generalization of n matrices abstract ky fan defines an n matrix to be a matrix of the form a ti b b 0 t b where b is the spectral radius of b and is the maximum of the spectral radii of all principal submatrices of b of order n 1 in this paper we define the closure n 0 matrices of n matrices by letting t it is shown that if a z and a 1 a n certain inequalities of n matrices are shown to hold for n 0 matrices and a method for constructing an n matrix from an m matrix is given

the reverse order law for the generalized inverse a 2 t s by using the ranks of matrices the necessary and sufficient condition is given for the reverse order law of the generalized inverse a 2 t s of matrix product a a 1a 2 a n to hold as the special cases similar results for moore penrose inverse a the weighted moore penrose inverse a m n drazin inverse a d and group inverse a g are also derived

newton s method and quasi newton sqp method for general lc1 constrained optimization in this paper we study the second order sufficient and necessary conditions for lc 1 constrained optimization problems present newton methods and quasi newton sqp methods for this kind of problems and establish locally linear and superlinear convergence this work extends the classical theory of convex smooth minimization to general lc 1 constrained optimization

the reverse order law for the generalized inverse at s 2 over a skew field abstract in this paper we study the reverse order law for the generalized inverse of the product of three matrices over a skew field with ranks of those matrices discussed and give a necessary and sufficient condition under which the reverse order law holds from this we obtain necessary and sufficient conditions for the reverse order laws for the known generalized inverses such as the moore penrose inverse the drazin inverse and the group inverse over skew fields

nonnegative matrices with power invariant zero patterns abstract if a is a nonsingular m matrix the elements of the sequence a k all have the same zero pattern using the drazin inverse we show that a similar zero pattern invariance property holds for a class of matrices which is larger than the generalized m matrices

periodic subvarieties of semiabelian varieties and annihilators of irreducible representations abstract let g be a semiabelian variety defined over a field of characteristic 0 endowed with an endomorphism we prove there is no proper subvariety y g which intersects the orbit of each periodic point of g under the action of as an application we are able to give a topological characterization of the annihilator ideals of irreducible representations in certain skew polynomial algebras

invariant varieties for polynomial dynamical systems we study algebraic dynamical systems and more generally s varieties f a n c a n c given by coordinatewise univariate polynomials by refining an old theorem of ritt on compositional identities amongst polynomials more precisely we find a nearly canonical way to write a polynomial as a composition of clusters from which one may easily read off possible compositional identities our main result is an explicit description of the weakly skew invariant varieties that is for a fixed field automorphism s c c those algebraic varieties x a n c for which f x x s as a special case we show that if f x c x is a polynomial of degree at least two that is not conjugate to a monomial chebyshev polynomial or a negative chebyshev polynomial and x a 2 c is an irreducible curve that is invariant under the action of x y f x f y and projects dominantly in both directions then x must be the graph of a polynomial that commutes with f under composition as consequences we deduce a variant of a conjecture of zhang on the existence of rational points with zariski dense forward orbits and a strong form of the dynamical manin mumford conjecture for liftings of the frobenius we also show that in models of acfa 0 a disintegrated set defined by s x f x for a polynomial f has morley rank one and is usually strongly minimal that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew conjugacy class of f is defined over a fixed field of a power of s and that nonorthogonality between two such sets is definable in families if the skew conjugacy class of f is defined over a fixed field of a power of s

reverse order laws for 1 2 3 generalized inverses in this paper we present purely algebraic necessary and sufficient conditions for reverse order laws to hold for 1 2 3 and 1 2 4 generalized inverses on the set of matrices

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

integer solution for linear complementarity problem we consider the problem of finding an integer solution to a linear complementarity problem we introduce the class i of matrices for which the corresponding linear complementarity problem has an integer complementary solution for every vector q for which it has a solution strong principal unimodularity forms a sufficient condition for inclusion in the class i it is also shown to be necessary for some well known classes of matrices including the class of positive semidefinite matrices this is used in deriving necessary and sufficient conditions for a convex quadratic program to have an integer optimal solution for all b and c for which it has an optimal solution characterizations are also derived for some other well known classes of matrices in linear complementarity to belong to the class i in the end a peeling algorithm for finding integer solution for linear complementarity problems is given and related results are derived

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

further results on the reverse order law for the moore penrose inverse in rings with involution abstract we present some equivalent conditions of the reverse order law for the moore penrose inverse in rings with involution extending some well known results to more general settings then we apply this result to obtain a set of equivalent conditions to the reverse order rule for the weighted moore penrose inverse in c algebras

characterizations of ep matrices and weighted ep matrices abstract a square complex matrix a is said to be ep if a and its conjugate transpose a have the same range in this paper we first collect a group of known characterizations of ep matrix and give some new characterizations of ep matrices then we define weighted ep matrix and present a wealth of characterizations for weighted ep matrix through various rank formulas for matrices and their generalized inverses

reverse order laws in c algebras abstract in this paper we offer purely algebraic necessary and sufficient conditions for reverse order laws for generalized inverses in c algebras extending rank conditions for matrices and range conditions for hilbert space operators

on hidden z matrices and the linear complementarity problem abstract in this paper we explore various matrix theoretic aspects of the hidden z class and demonstrate how the concept of principal pivot transform can be effectively used to extend many existing results in fact we revisit various results obtained for hidden z class by mangasarian cottle and pang in context of solving linear complementarity problems as linear programs we identify hidden z matrices of special category and discuss the number of solutions of the associated linear complementarity problems we also present game theoretic interpretation of various results related to hidden z class and obtain proofs following the game theoretic approach of raghavan for a subclass of z matrices

multiplicative perturbations of matrices and the generalized triple reverse order law for the moore penrose inverse abstract for a matrix a let a denote its moore penrose inverse a matrix m is called a multiplicative perturbation of t c m n if m e t f for some e c m m and f c n n based on the alternative expression for m as m e t t t f t t the generalized triple reverse order law for the moore penrose inverse is obtained as m f t t y y t z z l r 1 e t t where y y t z z l r 1 is the weighted moore penrose inverse for certain matrices y z l and r associated to the triple t e f furthermore it is proved that this weighted moore penrose inverse in the resulting expression for m can be really replaced with t if e t t e t t t t f t t f t t in the special case that rank m rank t or m is a weak perturbation of t a simplified version of m as well as m m and m m is also derived

on weak r monotonicity abstract in this paper the relationship between weak r monotonicity and 1 monotonicity is discussed in particular an affirmative answer to a question raised by werner in 1977 is given

generalized inverse of linear transformations a geometric approach abstract a generalized inverse of a linear transformation a where and are arbitrary finite dimensional vector spaces is defined using only geometrical concepts of linear transformations the inverse is uniquely defined in terms of specified subspaces l m and a linear transformation n satisfying some conditions such an inverse is called the lm n inverse a moore penrose type inverse is obtained by choosing n 0 some optimization problems are considered by choosing and as inner product spaces our results extend without any major modification of proofs to bounded linear operators with closed range on hilbert spaces

a note on the reverse order law for reflexive generalized inverse of multiple matrix products the reverse order law for reflexive generalized inverses of multiple matrix products was introduced and discussed in m wei reverse order laws for generalized inverse of multiple matrix products linear algebra appl 293 1999 273 288 there the author derived some necessary and sufficient conditions for the reverse order lawa n 1 2 a n 1 1 2 a 1 1 2 a 1a 2 a n 1 2 by applying the product singular value decomposition p svd in this note we revisited this reverse order law by using the extremal rank relations of generalized schur complements and a new simpler equivalent condition is obtained in terms of only the ranks of the known matrices

an invariance property related to the reverse order law abstract it is shown that quite surprisingly all matrices of the form l m where l and m denote generalized inverses of l and m are generalized inverses of ml if and only if the product mll m ml is invariant with respect to the choice of l and m which at the first glance looks to be a weaker condition than the requirement that mll m ml ml for every l and m this statement follows as an immediate corollary to the main result of the present note which provides two criteria for the invariance of expressions of the type kl m n involving four given matrices k l m n with generalized inverses l m of two of them

reverse order law for the group inverses this paper is to present some equivalent conditions concerning the reverse order law ab b a for the group invertible operators a b on a hilbert space h

superfluous matrices in linear complementarity superfluous matrices were introduced by howe 1983 in linear complementarity in general producing examples of this class is tedious a few examples can be found in chapter 6 of cottle pang and stone 1992 to overcome this problem we define a new class of matrices bar z and establish that in bar z superfluous matrices of any ordern 4 can easily be constructed for every integerk an example of a superfluous matrix of degreek is exhibited in the end

applications of completions of operator matrices to reverse order law for 1 inverses of operators on hilbert spaces abstract although the reverse order law for 1 generalized inverses of matrices has been completely resolved in this paper we do it for the first time for operators acting on separable hilbert spaces the approach taken here involves some of the previous research on completions of operator matrices to left and right invertibility we thus demonstrate usability of some of the results of that type by showing how they can be applied to one of the topics in generalized inverses of operators that has seen a great interest over the years

some notes on z matrices abstract an inequality is obtained relating the eigenvalue of minimum modulus of an n0 matrix a to the eigenvalue of minimum modulus of a a11 the schur complement of a11 in a where a11 is an m matrix similar results are obtained for irreducible m matrices and irreducible nonsingular f0 matrices circulant n0 matrices of order three are characterized in terms of their spectrum matrices whose inverses are f0 matrices are shown to have nonpositive almost principal minors

the degree of an exact order matrix the classes of exact order k matrices for any positive integer k were defined and studied by mohan parthasarathy and sridhar mohan s r t parthasarathy r sridhar 1994 the linear complementarity problem with exact order matrices math oper res 19 618 644 here we prove results on the linear complementarity problem lcpq m for m belonging to the class of exact order k k 3 using the concepts of degree theory our main result in this paper consists in proving that a matrix m rn n of exact order k for any positive integer n k 3 belongs to the class q if and only if the degree of m is either 1 or 1 also a complete characterization of exact order 2 matrices is presented in terms of their inverse structure

reverse order laws for least squares g inverses and minimum norm g inverses of products of two matrices in this paper we derive equivalent conditions for reverse order laws of least squares g inverses and minimum norm g inverses of product of two matrices a and b by applying the product singular value decomposition p svd

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

the absorption laws for the generalized inverses abstract in this paper we give necessary and sufficient conditions for the absorption laws in terms of 1 1 2 1 3 and 1 4 inverses also we consider the various types of mixed absorption law for the generalized inverses

equivalent conditions for generalized inverses of products abstract the relationships between generalized inverses of the product of two matrices a b and the product of generalized inverses of a b have been studied in the literature and equivalent conditions for b 1 a 1 ab 1 have been reported by applying the product singular value decomposition we derive more equivalent conditions for b 1 a 1 ab 1 and derive equivalent conditions for b 1 a 1 ab 1

applications of degree theory to linear complementarity problems in this paper we consider two applications of degree theory to linear complementarity problems in the first application we study the stability of an lcp at a solution point specifically we prove the stability of an lcp corresponding to a p0 matrix at an isolated solution using a recent degree formula due to stewart 1991 we strengthen a stability result of gowda and pang 1992 in the second application we use the same degree formula of stewart to describe the number of solutions of lcpm q when m is a negative almost n matrix this analysis leads to a lipschitzian characterization of the solution map i q solm q corresponding to a nondegenerate negative matrix

reverse order law of group inverses of products of two matrices we obtained several equivalent conditions for reverse order law of group inverses of product of two matrices several sufficient conditions which enable both ab b 1 2 a 1 2 and ba a 1 2 b 1 2 hold are also given

charakterisierungen von monotonen matrizen abstract it is well known that a matrix is of monotone kind if and only if it has a regular inverse all of whose entries are nonnegative motivated by a variety of practical and theoretical problems in the last 15 years several generalizations of that notion have appeared in the literature in this paper some generalizations of matrix monotonicity are investigated and practical characterizations are given

moore penrose inverse in rings with involution abstract we study the moore penrose inverse mp inverse in the setting of rings with involution the results include the relation between regular mp invertible and well supported elements we present an algebraic proof of the reverse order rule for the mp inverse valid under certain conditions on mp invertible elements applications to c algebras are given

when is b a a generalized inverse of ab in practice factorizations of a generalized inverse often arise from factorizations of the matrix which is to be inverted in addition to full rank factorizations normal factorizations and singular value decompositions svd there are other factorizations of particular matrices that are natural to certain problems for instance in statistics answers to questions like when is b a a generalized inverse of ab may thus be a computational tool besides they are of significant interest because they provide us with insights into the g inversion of matrix products

simultaneous kernels of matrix hadamard powers abstract in previous work belton et al 2016 2 the structure of the simultaneous kernels of hadamard powers of any positive semidefinite matrix was described key ingredients in the proof included a novel stratification of the cone of positive semidefinite matrices and a well known theorem of hershkowitz neumann and schneider which classifies the hermitian positive semidefinite matrices whose entries are 0 or 1 in modulus in this paper we show that each of these results extends to a larger class of matrices which we term 3 pmp principal minor positive

reverse order law for the moore penrose inverse abstract in this paper we present new results related to the reverse order law for the moore penrose inverse of operators on hilbert spaces some finite dimensional results are extended to infinite dimensional settings

reverse order laws for generalized inverses of multiple matrix products abstract in this article we study reverse order laws for generalized inverses and reflexive generalized inverses of the products of multiple matrices a 1 a n and the products of generalized inverses and reflexive generalized inverses of a n a 1 by applying the multiple product singular value decomposition we obtain necessary and sufficient conditions for one side inclusion relation of reverse order law for generalized inverses and necessary and sufficient conditions of reverse order law for reflexive generalized inverses

using rank formulas to characterize equalities for moore penrose inverses of matrix products a group of rank formulas related to the moore penrose inverses ab and b a are established various interesting consequences of these rank formulas are obtained the extensions of these results to the weighted moore penrose inverses ab m n and b p na m p are considered

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

the reverse order law revisited abstract necessary and sufficient conditions are given for the triple reverse order law abc c b a to hold some special cases are considered

density of orbits of endomorphisms of abelian varieties let a be an abelian variety defined over q and let be a dominant endomorphism of a as an algebraic variety we prove that either there exists a non constant rational fibration preserved by or there exists a point x a q whose orbit is zariski dense in a this provides a positive answer for abelian varieties of a question raised by medvedev and the second author in ms14 we prove also a stronger statement of this result in which is replaced by any commutative finitely generated monoid of dominant endomorphisms of a

on the maximum degree of an lcp map the linear complementarity problem can be shown to be equivalent to inverting a certain piecewise linear mapping we give bounds on the degree of this mapping and present a class of matrices that generates maps of high degree

the least squares g inverse for the sum of multiple matrices in this paper we give the necessary and sufficient conditions for the set identities a 1 a 2 a k 1 3 a 1 1 3 a 2 1 3 a k 1 3 and a 1 a 2 a k 1 4 a 1 1 4 a 2 1 4 a k 1 4 which concern the least squares g inverse 1 3 inverse and the minimum norm g inverse 1 4 inverse of sum of k matrices

a polynomial algorithm for testing the nonnegativity of principal minors of z matrices abstract a z matrix is a square matrix with nonpositive off diagonal elements we give a polynomial algorithm for testing the nonnegativity of principal minors of z matrices

mixed type reverse order laws of ab 1 2 3 and ab 1 2 4 abstract in this paper by using the extremal ranks of generalized schur complements we study the mixed type reverse order laws for ab 1 2 3 and ab 1 2 4 and the sufficient and necessary conditions are established for these two reverse order laws finally two numerical examples are given to illustrate our results

a polynomial time test for m matrices abstract a real square matrix with nonpositive off diagonal elements is a z matrix and a z matrix with nonnegative principal minors is an m matrix using principal pivoting we derive a polynomial time algorithm for testing whether a z matrix is an m matrix our algorithm is much simpler than a recent one proposed by ramamurthy we also study the zero nonzero structure of the inverse of a nonsingular m matrix and compare a 1 and b 1 in the case that a and b are nonsingular m matrices and b a

reverse order laws for the generalized inverses of multiple matrix products by using the ranks of matrices this article gives necessary and sufficient conditions for the reverse product a n a 2a 1of the moore penrose inverses of a1 a2 an to a 1 inverse inner inverse 1 2 inverse reflexive inner inverse 1 3 inverse 1 4 inverse 1 2 3 inverse 1 2 4 inverse or moore penrose inverse of matrix product a a1a2 an

perturbations of m matrices via abs methods and their applications to input output analysis some properties of m matrices are investigated in order to analyze the closure of the class with respect to perturbations introduced in the matrix an algorithm is derived by means of the implicit gauss cholesky algorithm in the abs class for linear systems a lower estimation of the solution of linear system ax b where a is an m matrix and the solutions of the leading principal subsystems are available at each iteration step for a total number of multiplications of order o n 33 the algorithm also tests whether a z matrix is an m matrix and the effects of changes in the entries of the coefficient matrix during the execution are investigated as well the number of the extra multiplications in general is less than those needed by the gaussian elimination finally an application to the open leontief static input output i o model is presented together with numerical results on a real case using the ibm 3090 200vf computer

mixed type reverse order law for products of three operators abstract we present new results related to the mixed type reverse order law for the moore penrose inverse of various products of three operators on hilbert spaces some finite dimensional results are extended to infinite dimensional settings

principal pivot transforms of some classes of matrices in this paper we study and characterize various classes of matrices that are defined based on principal pivot transforms we show that matrices in these classes have nonnegative principal minors

some relations of projection and star order in hilbert space abstract we study the star order on b h and show several properties for which the common star lower or star upper bound exists regarding the relationships among operators and projections we give various characterizations of star order and obtain much simpler matrix representations with respect to star order relation

triple reverse order law for weighted generalized inverses in this paper we present a new triple reverse order law of weighted generalized inverses which does not involve the calculation of any generalized inverse our result is an important improvement of the existing reverse order law of weighted generalized inverse

reverse order law in c algebras we study equivalent conditions for the reverse order law a1a2 an an a1 a1a2 anan a1 in c algebras as corollaries we obtain some recent and special results

a unified approach to one parametric general quadratic programming a method is proposed for finding local minima to the parametric general quadratic programming problem where all the coefficients are linear or polynomial functions of a scalar parameter the local minimum vector and the local minimum value are determined explicitly as rational functions of the parameter a numerical example is given

the representation and approximation for the weighted moore penrose inverse we present a unified representation theorem for the weighted moore penrose inverse specific expressions and computational procedures for the weighted moore penrose inverse can be uniformly derived

forward order law for g inverses of the product of two matrices abstract in this paper we derive equivalent conditions for a b ab by the method of maximal rank and thus we obtain equivalent conditions for the forward order laws of the product of two matrices

reverse order laws for the drazin inverses abstract this paper is to present some equivalent conditions concerning the reverse order law p q d q d p d for the drazin invertible operators p and q with partial commutative properties p p q 0 or q p q 0 or p p q q d 0 or q p p d q 0

mixed type reverse order laws of ab 1 3 ab 1 2 3 and ab 1 3 4 in this paper by using the extremal ranks of generalized schur complements we study the mixed type reverse order laws for ab 1 3 ab 1 2 3 and ab 1 3 4 and the sufficient and necessary conditions are established for these reverse order laws

mixed type reverse order law of ab 13 some mixed type reverse order laws of ab have been proposed and studied by tian in this paper by applying the extremal rank relations of generalized schur complements and the p svd of two matrices a and b we study two kind of mixed type reverse order laws for ab 13 obtain necessary and sufficient conditions for these reverse order laws