On Hermitian nonnegative-definite solutions to matrix equations

Liu, Xu-Qing; Rong, Jian-Ying

doi:10.1134/s000143460903016x

Cited by 2 publications

(3 citation statements)

References 5 publications

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“…✷ Corollary 3.6(e)-(l) give a set of analytical characterizations for the existence of definite common solutions of the two matrix equations in (3.23) by using some rank and range equalities and inequalities. These characterizations are simple and easy to understand in comparison with various known conditions (see, e.g., [14,40,41])s on the existence of definite common solutions of (3.23).…”

Section: ) ✷mentioning

confidence: 93%

Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions

Tian¹

2014

Banach J. Math. Anal.

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Matrix rank and inertia optimization problems are a class of discontinuous optimization problems in which the decision variables are matrices running over certain matrix sets, while the ranks and inertias of the variable matrices are taken as integer-valued objective functions. In this paper, we establish a group of explicit formulas for calculating the maximal and minimal values of the rank and inertia objective functions of the Hermitian matrix expression A1 − B1XB * 1 subject to the common Hermitian solution of a pair of consistent matrix equations B2XB * 2 = A2 and B3XB * 3 = A3, and Hermitian solution of the consistent matrix equation B4X = A4, respectively. Many consequences are obtained, in particular, necessary and sufficient conditions are established for the triple matrix equations B1XB * 1 = A1, B2XB * 2 = A2 and B3XB * 3 = A3 to have a common Hermitian solution, as necessary and sufficient conditions for the two matrix equations B1XB * 1 = A1 and B4X = A4 to have a common Hermitian solution. are given, i = 1, 2, 3, and X ∈ C n H is a variable matrix, and study the following constrained optimization problems: Problem 1.1 For the constrained linear matrix-valued function in (1.2), establish explicit formulas for calculating max X∈C n H r(

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Section: ) ✷mentioning

confidence: 93%

Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions

Tian¹

2014

Banach J. Math. Anal.

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“…For example, the coupled Sylvester matrix equations often arise in computing an additive decomposition of a generalized transform matrix equations, in computing deflated subspaces of descriptor linear systems, and in problems of reordering eigenvalues of regular matrix pairs. The literature on the approximation of solutions of linear matrix equations is large and is still growing rapidly 13‐22 . By applying the Moore–Penrose generalized inverse and the Kronecker product of matrices, Yuan and Liao proposed the expressions of various least squares solutions with the least norm of the quaternion matrix equation 23 .…”

Section: Introductionmentioning

confidence: 99%

“…The literature on the approximation of solutions of linear matrix equations is large and is still growing rapidly. [13][14][15][16][17][18][19][20][21][22] By applying the Moore-Penrose generalized inverse and the Kronecker product of matrices, Yuan and Liao proposed the expressions of various least squares solutions with the least norm of the quaternion matrix equation. 23 Chiang presented some useful sufficient conditions for the solvability of the linear matrix equation AX + (X)B = C and derived the expressions of the explicit solutions of the matrix equation.…”

Section: Introductionmentioning

confidence: 99%

Least‐squares partially bisymmetric solutions of coupled Sylvester matrix equations accompanied by a prescribed submatrix constraint

Hajarian

Chronopoulos

2020

Math Methods in App Sciences

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The coupled Sylvester matrix equations (CSMEs) A1XB1+C1YD1=E1,A2XB2+C2YD2=E2, appear frequently in various fields of mathematics and engineering such as in control systems and signal processing. In this investigation, we establish and analyze the generalized conjugate directions (GCDs) method for solving the CSMEs over partially bisymmetric matrices X and Y with a prescribed submatrix constraint. We show that the GCD method with the arbitrary initial bisymmetric matrices can compute the least‐squares partially bisymmetric solutions with a prescribed submatrix constraint within a finite number of iterations in the absence of round‐off errors. Numerical examples illustrate the efficiency and simplicity of the GCD method and confirm the theoretical results.

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On Hermitian nonnegative-definite solutions to matrix equations

Cited by 2 publications

References 5 publications

Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions

Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions

Least‐squares partially bisymmetric solutions of coupled Sylvester matrix equations accompanied by a prescribed submatrix constraint

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