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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

##### MSC:
 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
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