Symmetric solutions of linear matrix equations by matrix decompositions
type: Item
instance of: scholarly article
title: Symmetric solutions of linear matrix equations by matrix decompositions
zbMATH Open document ID: 0688.15003
DOI: 10.1016/0024-3795(89)90067-0
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: This paper considers the symmetric solutions of linear matrix equations (1) \(AX=B.\) When the matrix A is in its singular-value decomposition, (1) becomes \(UDV^ TX=[U_ 1,U_ 2]\left[ \begin{matrix} \Sigma \\ 0\end{matrix} \begin{matrix} 0\\ 0\end{matrix} \right]\left[ \begin{matrix} V^ T_ 1\\ V^ T_ 2\end{matrix} \right]X=B,\) where the matrices U and V are orthogonal and \(\Sigma\) is positively diagonal. It is shown that \(U^ T_ 2B\) has to be a zero matrix for consistency and \(asym(\Sigma^{-1}U^ T_ 1BV_ 1)\) has to be a zero matrix for symmetry, where \(asym(P)=(P-P^ T)/2.\) In Section 3, when the consistency and symmetry conditions are not satisfied, the least-squares symmetric solutions of equation (1) are considered. Numerically, it will be difficult to check \(U^ T_ 2B=0\) and \(asym(U^ T_ 1BV_ 1)=0\). In Section 4, the problem is replaced by the need to calculate the generalized singular-value decomposition. In Section 5, symmetric and nonsymmetric solutions of other linear matrix equations \(A^ TXB=C,\quad AX^ T+X^ TA=C,\quad A^ TXB+B^ TX^ TA=C,\quad [AX,CX]=[B,D]\) are looked at, using the singular-value and generalized singular-value decompositions as well as the real Schur and real generalized Schur decompositions. The related numerical algorithms are discussed briefly when appropriate.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A23
zbMATH DE Number: 4127414
zbMATH Keywords: matrix decompositions
zbMATH Keywords: symmetric solutions
zbMATH Keywords: linear matrix equations
zbMATH Keywords: least-squares symmetric solutions
zbMATH Keywords: singular-value decomposition
zbMATH Keywords: Schur decompositions
zbMATH Keywords: numerical algorithms
author: King-wah Eric Chu
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90067-0
OpenAlex ID: W1976329209
cites work: Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4]
cites work: Singular value and generalized singular value decompositions and the solution of linear matrix equations
cites work: The solution of the matrix equations \(AXB-CXD=E\) and \((YA-DZ,YC- BZ)=(E,F)\)
cites work: Norm-Preserving Dilations and Their Applications to Optimal Error Bounds
cites work: On the symmetric solutions of a linear matrix equation
cites work: Q3254327
cites work: A Hessenberg-Schur method for the problem AX + XB= C
cites work: Q5185900
cites work: Some matrix equations over a finite field
cites work: Explicit Solutions of Linear Matrix Equations
cites work: L-structured matrices and linear matrix equations∗
cites work: The commutation matrix: Some properties and applications
cites work: The Elimination Matrix: Some Lemmas and Applications
cites work: Q4115244
cites work: Towards a Generalized Singular Value Decomposition
cites work: Computing the CS decomposition of a partitioned orthonormal matrix
cites work: Vector structures and solutions of linear matrix equations
description: scientific article; zbMATH DE number 4127414