second order directional derivatives of spectral functions a spectral function of a symmetric matrix x is a function which depends only on the eigenvalues of x l 1 x l 2 x l n x and may be written as l 1 x l 2 x l n x for some symmetric function in this paper we assume that is a c 1 1 function and discuss second order directional derivatives of such a spectral function we obtain an explicit expression of second order directional derivative for the spectral function

closed form solutions to the nonhomogeneous yakubovich conjugate matrix equation some closed form solutions are provided for the nonhomogeneous yakubovich conjugate matrix equation email protected f by r with x and y being unknown matrices the presented solutions can offer all the degrees of freedom which is represented by an arbitrarily chosen parameter matrix the primary feature of the solutions is that the matrices f and r are not restricted to be in any canonical form or may be even unknown a priori one of the solutions is neatly expressed in terms of controllability matrices and observability matrices

singular value and generalized singular value decompositions and the solution of linear matrix equations abstract the solution of the linear matrix equations i axb cyd e and ii axb fxg e h are considered new necessary and sufficient conditions for the consistency of the equations are derived some using the generalized singular value decomposition special cases iii ax yd e and iv axb e are treated using the singular value decomposition numerical algorithms for the solutions are also suggested

a pair of simultaneous linear matrix equations a1xb1 c1 a2xb2 c2 and a matrix programming problem necessary and sufficient conditions are obtained for a pair of matrix equations a1xb1 c1 a2xb2 c2 on a general field f to have a common solution along with the expression for a general common solution when certain conditions hold common solutions of minimum rank are described a matrix programming problem is solved en route

finite iterative algorithms for a common solution to a group of complex matrix equations abstract an iterative algorithm is constructed to give a common solution to a group of complex matrix equations by using the proposed algorithm the existence of a common solution can be determined automatically when a common solution exists for this group of matrix equations it is proven by using a real inner product in complex matrix spaces as a tool that a solution can be obtained within finite iteration steps for any initial values in the absence of round off errors the algorithm is also generalized to solve a more general case a numerical example is given to illustrate the effectiveness of the proposed method

a finite iterative method for solving a pair of linear matrix equations axb cxd e f abstract in this paper an efficient iterative method is presented to solve a pair of linear matrix equations axb cxd e f with real matrix x by this iterative method the solvability of the matrix equations pair can be determined automatically when the pair of matrix equations are consistent then for any initial matrix x0 a solution can be obtained within finite iteration steps in the absence of roundoff errors and the least norm solution can be obtained by choosing a special kind of initial matrix in addition the unique optimal approximation solution x to a given matrix x in frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations a x b c x d e f where e e a x b f f c x d the given numerical examples demonstrate that the iterative method is quite efficient

spectral bundle methods for non convex maximum eigenvalue functions second order methods we study constrained and unconstrained optimization programs for nonconvex maximum eigenvalue functions we show how second order techniques may be introduced as soon as it is possible to reliably guess the multiplicity of the maximum eigenvalue at a limit point we examine in which way standard and projected newton steps may be combined with a nonsmooth first order method to obtain a globally convergent algorithm with a fair chance to local superlinear or quadratic convergence

partially augmented lagrangian method for matrix inequality constraints we discuss a partially augmented lagrangian method for optimization programs with matrix inequality constraints a global convergence result is obtained applications to hard problems in feedback control are presented to validate the method numerically

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 matrix equations ax b xc d with px sxp constraint abstract the matrix equations ax b xc d with the constraint px sxp are considered where p is a given hermitian matrix satisfying p 2 i and s 1 using an eigenvalue decomposition of p the constrained problem can be equivalently transformed to two independent unconstrained problems of matrix equations in similar forms and hence the constrained problem can be solved in terms of the eigenvectors of p in this paper a simple and eigenvector free formula of the general solutions to the constrained problem is presented using moore penrose generalized inverses of the coefficient matrices a and c a similar problem of the matrix equations with generalized constraint is discussed too we also consider the least norm solution of the constrained problems

first order sensitivity analysis of symplectic eigenvalues abstract for every 2 n 2 n positive definite matrix a there are n positive numbers d 1 a d n a associated with a called the symplectic eigenvalues of a it is known that d m are continuous functions of a but are not differentiable in general in this paper we show that the directional derivative of d m exists and derive its expression we also discuss various subdifferential properties of d m such as clarke and michel penot subdifferentials

generalizations of cliques odd cycles and anticycles and their relation to independence system polyhedra padberg padberg m w 1973 on the facial structure of set packing polyhedra math programming5 199 215 has shown that if a graph g with vertex set v constitutes a clique odd cycle odd anticycle then the inequality x vxv 1 x vxv v 1 x vxv 2 defines a facet of pistg the convex hull of the incidence vectors of stable sets in g we generalize the concepts of cliques odd cycles and anticycles to arbitrary independence systems e i and show that the corresponding inequalities define facets of pi the convex hull of the incidence vectors of independent sets

the matrix nearness problem for symmetric matrices associated with the matrix equation atxa btxb c d abstract a direct method based on the projection theorem in inner products spaces the generalized singular value decomposition and the canonical correlation decomposition is presented for finding the optimal approximate solution x in the set s e to a given matrix x where s e denotes the least squares symmetric solution set of the matrix equation atxa btxb c d the analytical expression of the optimal approximate solution x is obtained and an algorithm for finding this solution is also suggested

ranks of least squares solutions of the matrix equation axb c for a complex matrix equation axb c we solve the following two problems 1 the maximal and minimal ranks of least square solution x to axb c and 2 the maximal and minimal ranks of two real matrices x 0 and x 1 in least square solution x x 0 ix 1 to axb c we also give a necessary and sufficient condition for matrix equations a ix ib i c i i 1 2 to have a common least square solution

sensitivity analysis of all eigenvalues of a symmetric matrix given a x a ij x a n by n symmetric matrix depending smoothly on a parameter x we study the first order sensitivity of all the eigenvalues lambda m x of a x 1 le m le n under some smoothness assumption like the a ij be c 1 we prove that the directional derivatives d mapsto lambda prime m x d lim t to o lambda m x td lambda m x t do exist and give an explicit expression of them in terms of the data of the parametrized matrix the key idea to circumvent the difficulties inherent to the study of each lambda m taken separately is to consider the functions f m x 1 le m le n defined as the sums of the m largest eigenvalues of a x based on ky fan s variational formulation of f m and some chain rule from nonsmooth analysis we derive an explicit formula for the generalized gradient of f m and a computationally useful formula for the directional derivative of f m using these formulas and the relation lambda m f m f m 1 we then derive the directional derivative of lambda m some properties of this directional derivative as well as an illustrative example are presented

nearest southeast submatrix that makes multiple a prescribed eigenvalue part 1 abstract given four complex matrices a b c and d where a c n n and d c m m and given a complex number z 0 what is the spectral norm distance from d to the set of matrices x c m m such that z 0 is a multiple eigenvalue of the matrix a b c x this problem is solved when z 0 is not an eigenvalue of a we also give a conjecture when z 0 is an eigenvalue of a

large scale optimization of eigenvalues optimization problems involving eigenvalues arise in many applications let x be a vector of real parameters and let a x be a continuously differentiable symmetric matrix function of x we consi

on the analysis of the discretized kohn sham density functional theory in this paper we study a few theoretical issues in the discretized kohn sham ks density functional theory the equivalence between either a local or global minimizer of the ks total energy minimization problem and the solution to the ks equation is established under certain assumptions the nonzero charge densities of a strong local minimizer are shown to be bounded from below by a positive constant uniformly we analyze the self consistent field scf iteration by formulating the ks equation as a fixed point map with respect to the potential the jacobian of these fixed point maps is derived explicitly both global and local convergence of the simple mixing scheme can be established if the gap between the occupied states and unoccupied states is sufficiently large this assumption can be relaxed in certain cases numerical experiments based on the matlab toolbox kssolv show that it holds on a few simple examples although our assumption on the gap is very stringent our analysis is still valuable for

spectral operators of matrices the class of matrix optimization problems mops has been recognized in recent years to be a powerful tool by researchers far beyond the optimization community to model many important ap plications involving structured low rank matrices this trend can be credited to some extent to the exciting developments in the emerging field of compressed sensing the lowner operator which gen erates a matrix valued function by applying a single variable function to each of the singular values of a matrix has played an important role for a long time in solving matrix optimization problems however the classical theory developed for lowner operators has become inadequate in these recent applications the main objective of this paper is to provide some necessary theoretical foundations for designing numerical methods for solving the mop this goal is achieved by introducing and conducting a thorough study on a new class of matrix valued functions coined as spectral operators of matrices several fundamental properties of spectral operators including the well definedness continuity di rectional differentiability frechet differentiability locally lipschitzian continuity order b ouligand differentiability 0 1 order g semismooth 0 1 and the characterization of clarke s generalized jacobian are systematically studied

quadratic expansions of spectral functions a function f on the space of n n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument that is f a f uau t for every orthogonal u and symmetric a in its domain spectral functions are in one to one correspondence with the symmetric functions on r n those that are invariant under arbitrary swapping of their arguments in this paper we show that a spectral function has a quadratic expansion around a point a if and only if its corresponding symmetric function has quadratic expansion around a the vector of eigenvalues we also give a concise and easy to use formula for the hessian of the spectral function in the case of convex functions we show that a positive definite hessian of f implies positive definiteness of the hessian of f 2002 elsevier science inc all rights reserved

an efficient algorithm for the least squares reflexive solution of the matrix equation a1xb1 c1 a2xb2 c2 abstract a n n real matrix p is said to be a real generalized reflection matrix if pt p and p2 i a n n real matrix x is said to be a reflexive matrix with respect to the generalized reflection matrix p if x pxp in this paper the concept of gradient matrix f x is presented and an algorithm is constructed to solve the reflexive with respect to the generalized reflection matrix p solution of the minimum frobenius norm residual problem a 1 xb 1 a 2 xb 2 c 1 c 2 min by this algorithm for any initial reflexive matrix x1 a solution x can be obtained within finite iteration steps in the absence of roundoff errors and the solution x with least norm can be obtained by choosing a special kind of initial reflexive matrix in addition in the solution set of above problem the unique optimal approximation solution x to a given matrix x0 in frobenius norm can be obtained by finding the least norm reflexive solution x of the new minimum residual problem min a 1 x b 1 a 2 x b 2 c 1 c 2 where c 1 c 1 a 1 x 0 b 1 c 2 c 2 a 2 x 0 b 2 given numerical examples show that the iterative method is quite efficient

on solutions of the matrix equations x axb c and x axb c this paper studies the solutions of complex matrix equations x axb c and x axb c and obtains explicit solutions of the equations by the method of characteristic polynomial and a method of real representation of a complex matrix respectively

asymptotic expansions of the ordered spectrum of symmetric matrices abstract in this work we build on ideas of torki 2001 6 and show that if a symmetric matrix valued map t a t has a one sided asymptotic expansion at t 0 of order k then so does t m a t where m is the m th largest eigenvalue we derive formulas for computing the coefficients a 0 a 1 a k in the asymptotic expansion as an application of the approach we give a new proof of a classical result due to kato 1976 3 about the one sided analyticity of the ordered spectrum under analytic perturbations finally as a demonstration of the derived formulas we compute the first three terms in the asymptotic expansion of m a t e for any fixed symmetric matrices a and e

an introduction to a class of matrix cone programming in this paper we define a class of linear conic programming which we call matrix cone programming or mcp involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone mcp has recently been found to have many important applications for example in nuclear norm relaxations of affine rank minimization problems in order to make the defined mcp tractable and meaningful we must first understand the structure of these epigraphs so far only the epigraph of the frobenius matrix norm which can be regarded as a second order cone has been well studied here we take an initial step to study several important properties including its closed form solution calm bouligand differentiability and strong semismoothness of the metric projection operator over the epigraph of the l 1 l infty spectral or operator and nuclear matrix norm respectively these properties make it possible to apply augmented lagrangian methods which have recently received a great deal of interests due to their high efficiency in solving large scale semidefinite programming to this class of mcp problems the work done in this paper is far from comprehensive rather it is intended as a starting point to call for more insightful research on mcp so that it can serve as a basic tool to solve more challenging convex matrix optimization problems in years to come

an iterative algorithm for solving a pair of matrix equations ayb e cyd f over generalized centro symmetric matrices a matrix email protected r n x n is said to be a symmetric orthogonal matrix if p p t p 1 a matrix email protected r n x n is said to be generalized centro symmetric generalized central anti symmetric with respect to p if a pap a pap the generalized centro symmetric matrices have wide applications in information theory linear estimate theory and numerical analysis in this paper we propose a new iterative algorithm to compute a generalized centro symmetric solution of the linear matrix equations ayb e cyd f we show when the matrix equations are consistent over generalized centro symmetric matrix y for any initial generalized centro symmetric matrix y 1 the sequence y k generated by the introduced algorithm converges to a generalized centro symmetric solution of matrix equations ayb e cyd f the least frobenius norm generalized centro symmetric solution can be derived when a special initial generalized centro symmetric matrix is chosen furthermore the optimal approximation generalized centro symmetric solution to a given generalized centro symmetric matrix can be derived several numerical examples are given to show the efficiency of the presented method

least squares solution with the minimum norm for the matrix equation axb gxh c d based on the projection theorem in hilbert space by making use of the generalized singular value decomposition and the canonical correlation decomposition an analytical expression of the least squares solution for the matrix equation axb gxh c d with the minimum norm is derived an algorithm for finding the minimum norm solution is described and some numerical results have been given