A unified treatment for the restricted solutions of the matrix equation $AXB=C$

Xu, Jing; Zhang, Hairui; Liu, Lina; Zhang, Huiting; Yuan, Yongxin

doi:10.3934/math.2020424

Cited by 2 publications

(1 citation statement)

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“…An equation as such was first proposed and studied by Sir Penrose in his seminal paper [6], and he obtained a group of exact and analytical results concerning the solvability condition and the general solution of the equation using the ranks, the ranges, and the generalized inverses of the given matrices in the equation. These results were recognized as the fundamental and classic materials in the research field of various linear matrix equations in the 1950s, and extensively prompted many deep-going investigations of linear matrix equations from theoretical and computational points of view in the past several decades; see [6][7][8][9][10][11][12][13][14] for several earlier and recent papers regarding this subject. Below, we describe our motivation in the study of (1).…”

Section: Introductionmentioning

confidence: 99%

On Relationships between a Linear Matrix Equation and Its Four Reduced Equations

Jiang

Tian

Yuan

2022

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Given the linear matrix equation AXB=C, we partition it into the form A1X11B1+A1X12B2+A2X21B1+A2X22B2=C, and then pre- and post-multiply both sides of the equation by the four orthogonal projectors generated from the coefficient matrices A1, A1, B1, and B2 to obtain four reduced linear matrix equations. In this situation, each of the four reduced equations involves just one of the four unknown submatrices X11, X12, X21, and X22, respectively. In this paper, we study the relationships between the general solution of AXB=C and the general solutions of the four reduced equations using some highly selective matrix analysis tools in relation to ranks, ranges, and generalized inverses of matrices.

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Section: Introductionmentioning

confidence: 99%

On Relationships between a Linear Matrix Equation and Its Four Reduced Equations

Jiang

Tian

Yuan

2022

Axioms

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The Hermitian solution to a matrix inequality under linear constraint

Chen,

Duan

2024

MATH

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<abstract><p>In this paper, the necessary and sufficient conditions under which the matrix inequality $ C^*XC\geq D\ (>D) $ subject to the linear constraint $ A^*XA = B $ is solvable are deduced by means of the spectral decompositions of some matrices and the generalized singular value decomposition of a matrix pair. An explicit expression of the general Hermitian solution is also provided. One numerical example demonstrates the effectiveness of the proposed method.</p></abstract>

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A unified treatment for the restricted solutions of the matrix equation $AXB=C$

Cited by 2 publications

References 13 publications

On Relationships between a Linear Matrix Equation and Its Four Reduced Equations

On Relationships between a Linear Matrix Equation and Its Four Reduced Equations

The Hermitian solution to a matrix inequality under linear constraint

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