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Symmetric solutions of linear matrix equations by matrix decompositions. (English) Zbl 0688.15003
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.
Reviewer: M.Kono

15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A23 Factorization of matrices
Full Text: DOI
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