Hermitian Solutions Research Articles

The general solution to a system of adjointable operator equations over Hilbert C^∗-modules

We establish necessary and sufficient conditions for the existence of solution to the system of adjointable operator equations A1X = D1,XB2 = D2,A3XB3 +B3X ∗C3 = D3 over the Hilbert C∗ -modules. We also give the explicit expression of the general solution to this system when the solvability conditions are satisfied. As an application, we investigate the anti-reflexive Hermitian solution to the system of complex matrix equations AX = B,XC = D,EXE∗ = F . The findings of this paper extend some known results in the literature. Mathematics subject classification (2010): 47A62, 46L08, 15A24, 47A52.

Solutions with special structure to the linear matrix equationAX=B

In this article we solve the following three kinds of problems. First, some formulas for the minimal rank of the submatrices in a solution X of matrix equation A X = B and the minimal and maximal rank of X itself are derived by using the matrix rank method. From these formulas, necessary and sufficient conditions are given for X to be nonsingular or the submatrices to be zero, respectively. Second, some formulas for the minimal rank of X + X ∗ , X − X ∗ and the corresponding expressions of submatrices of X are investigated. Combined with matrix decomposition, necessary and sufficient conditions are given for the existence of solutions to be Hermitian, local Hermitian, Skew-Hermitian and local Skew-Hermitian, respectively. Third, necessary and sufficient conditions are given for the existence of solutions to be local positive (negative) semidefinite, and for some Hermitian solution with zero submatrix, the structure form of a positive (negative) semidefinite solution are obtained using results of inertia.

On Additive Decomposition of the Hermitian Solution of the Matrix Equation AXA* = B

The decomposition of a Hermitian solution of the linear matrix equation AXA* = B into the sum of Hermitian solutions of other two linear matrix equations \({A_{1}X_{1}A^{*}_{1} = B_{1}}\) and \({A_{2}X_{2}A^*_{2} = B_{2}}\) are approached. As applications, the additive decomposition of Hermitian generalized inverse C − = A − + B − for three Hermitian matrices A, B and C is also considered.

Max-Min Problems on the Ranks and Inertias of the Matrix Expressions A−BXC±(BXC)∗ with Applications

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ∗), where A=±A ∗ and (⋅)∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ∗ subject to Hermitian solutions of a consistent matrix equation AXA ∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ∗ A ∼ B with respect to a Hermitian generalized inverse A ∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.

The matrix nearness problem associated with the quaternion matrix equation AXA H +BYB H =C

In this paper, by making use of the CCD-Q, GSVD-Q, and the projection theorem in the finite dimensional inner product space, we derive the expression of Hermitian solution for the matrix nearness problem associated with the quaternion matrix equation AXAH+BYBH=C.

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2 × 2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A - BXB ∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX = B , as well as the extremal inertias of a partial block Hermitian matrix.

Common Hermitian solutions to some operator equations on Hilber [formula omitted]-modules

We establish necessary and sufficient conditions for the existence of the general common Hermitian solution to the equations A 1 X = C 1 , XB 1 = C 2 , A 3 XA 3 ∗ = C 3 , A 4 XA 4 ∗ = C 4 for adjointable operators between Hilbert C ∗ -modules, and present an expression for the common Hermitian solution to the equations in terms of Moore–Penrose inverse of operators when the solvability conditions are satisfied. The findings of this paper extend some known results in the literature.

A simultaneous decomposition of a matrix triplet with applications

Researches on ranks of matrix expressions have posed a number of challenging questions, one of which is concerned with simultaneous decompositions of several given matrices. In this paper, we construct a simultaneous decomposition to a matrix triplet (A, B, C), where A=±A*. Through the simultaneous matrix decomposition, we derive a canonical form for the matrix expressions A−BXB*−CYC* and then solve two conjectures on the maximal and minimal possible ranks of A−BXB*−CYC* with respect to X=±X* and Y=±Y*. As an application, we derive a sufficient and necessary condition for the matrix equation BXB* + CYC*=A to have a pair of Hermitian solutions, and then give the general Hermitian solutions to the matrix equation. Copyright © 2010 John Wiley & Sons, Ltd.

Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation

The extreme ranks, i.e., the maximal and minimal ranks, are established for the general Hermitian solution as well as the general skew-Hermitian solution to the classical matrix equation AXA ¤ + BY B ¤ = C over the quaternion algebra. Also given in this paper are the formulas of extreme ranks of real matrices Xi, Yi, i = 1,··· , 4, in a pair (skew-)Hermitian solution X = X1 + X2i + X3j + X4k, Y = Y1 + Y2i + Y3j + Y4k. Moreover, the necessary and sufficient conditions for the existence of a real (skew-)symmetric solution, a complex (skew-)Hermitian solution, and a pure imaginary (skew-)Hermitian solution to the matrix equation mentioned above are presented in this paper. Also established are expressions of such solutions to the equation when corresponding solvability conditions are satisfied. The findings of this paper widely extend the known results in the literature.

Hermitian tridiagonal solution with the least norm to quaternionic least squares problem

Quaternionic least squares (QLS) is an efficient method for solving approximate problems in quaternionic quantum theory. In view of the extensive applications of Hermitian tridiagonal matrices in physics, in this paper we list some properties of basis matrices and subvectors related to tridiagonal matrices, and give an iterative algorithm for finding Hermitian tridiagonal solution with the least norm to the quaternionic least squares problem by making the best use of structure of real representation matrices, we also propose a preconditioning strategy for the Algorithm LSQR-Q in Wang, Wei and Feng (2008) [14] and our algorithm. Numerical experiments are provided to verify the effectiveness of our method.

Solutions to operator equations on Hilbert [formula omitted]-modules

Let A be a C ∗ -algebra, E , F and G be Hilbert A -modules, T ∈ L A ( E , F ) , and T ′ ∈ L A ( G , F ) . We generalize the Douglas theorem about the operator equation TX = T ′ from Hilbert space to Hilbert C ∗ -module. To the equation TX = T ′ and to the system of two equations TX = T ′ and XS = S ′ , we get the forms of general solutions (in the case that there exists a solution), and give some sufficient and necessary conditions for the existence of solutions, and the existence of hermitian solutions and positive solutions (in the case G = E ). In addition, the forms of general hermitian solution and general positive solution (in the case that there exists a solution and G = E ) to the equation TX = T ′ are given too.

Explicit polynomial formulas for solutions of the matrix equation AX−XA=C

We provide some mathematical tool to analyze the possible theoretical structure of Hamiltonian reconstructed from von Neumann or Heisenberg’s equation for a finite-dimensional quantum system. To this end, we give an explicit polynomial representation for the general solution of the matrix equation AX−XA=C together with some (also polynomial) solvability conditions when A and C are some complex square matrices of the same size and the matrix A is semisimple. When the matrix A is normal, we derive a polynomial existence condition for Hermitian solutions and a similar formula for all solutions of this type.

PERTURBATION ANALYSIS FOR THE MATRIX EQUATIONS X ± A∗X−1A = I

The nonlinear matrix equations $X\pm A^{*}X^{-1}A=I$ are investigated, where $A$ is an $n\times n$ nonsingular matrix and $I$ is an $n\times n$ identity matrix. Some new perturbation bounds for Hermitian positive definite solutions of these equations are derived by using elementary calculus techniques developed in[Sun J G , BIT, 31(1991), pp.341-352] and [Barrlund A, BIT, 31(1991), pp.358-363]. The new results are illustrated by numerical examples.

Ranks of Hermitian and skew-Hermitian solutions to the matrix equation [formula omitted

Given a complex matrix equation AXA ∗ = B , where B ∗ = ± B , we present explicit formulas for the maximal and minimal ranks of Hermitian (skew-Hermitian) solutions X to the equation as well as the maximal and minimal ranks of the real matrices X 0 and X 1 in a Hermitian (skew-Hermitian) solution X = X 0 + iX 1 . As applications, we give the maximal and minimal ranks of the real matrices C and D in a Hermitian (skew-Hermitian) g -inverse ( A + iB ) - = C + iD of a Hermitian (skew-Hermitian) matrix A + iB .

On Hermitian nonnegative-definite solutions to matrix equations

In this note, for a system of q matrix equations of the form $$ A_i XA_i^* = B_i B_i^* ,i = 1,2,...,q, $$ we consider the problem of the existence of Hermitian nonnegative-definite solutions. We offer an alternative with simplification and regularity to the result on necessary and sufficient conditions for the above matrix equations with q = 2 to have a Hermitian nonnegative-definite solution obtained by Zhang [1], who proposed a revised version of Young et al. [2]. Moreover, we give a necessary condition for the general case and then put forward a conjecture, with which at least some special situations are in agreement.

Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA *=B with applications

Suppose that AXA*=B is a consistent matrix equation and partition its Hermitian solution X*=X into a 2-by-2 block form. In this paper, we give some formulas for the maximal and minimal ranks of the submatrices in an Hermitian solution X to AXA*=B. From these formulas we derive necessary and sufficient conditions for the submatrices to be zero or to be unique, respectively. As applications, we give some properties of Hermitian generalized inverses for an Hermitian matrix.

Some investigation on Hermitian positive definite solutions of the matrix equation [formula omitted

In this paper, the Hermitian positive definite solutions of the matrix equation X s + A ∗ X - t A = Q are considered, where Q is an Hermitian positive definite matrix, s and t are positive integers. Necessary and sufficient conditions for the existence of an Hermitian positive definite solution are derived. A sufficient condition for the equation to have only two different Hermitian positive definite solutions and the formulas for these solutions are obtained. In particular, the equation with the case AQ 1 2 = Q 1 2 A is discussed. A necessary condition for the existence of an Hermitian positive definite solution and some new properties of the Hermitian positive definite solutions are given, which generalize the existing related results.

The Hermitian reflexive solutions to the matrix inverse problem [formula omitted

In this paper, the necessary and sufficient conditions for the solvability of the matrix inverse problem AX = B in Hermitian reflexive matrix set HJ n × n are given, and the solution expression of this matrix inverse problem is also presented. Moreover, an optimal approximation to any matrix in the above solution set in Frobenius norm is provided.

Necessary and Sufficient Conditions for the Existence of a Hermitian Positive Definite Solution of a Type of Nonlinear Matrix Equations

We study the Hermitian positive definite solutions of the nonlinear matrix equation X + A∗X−2A = I, where A is an n × n nonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations of X + A∗X−2A = I are presented while the matrix equation has a Hermitian positive definite solution.

On Hermitian positive definite solution of the matrix equation [formula omitted

A conjecture that the nonlinear matrix equation X − ∑ i = 1 m A i ∗ X r A i = Q ( − 1 ≤ r < 0 or 0 < r < 1 ) always has a unique Hermitian positive definite solution is proved. Some bounds of the unique Hermitian positive definite solution are given.