Symmetric solutions of linear matrix equations by matrix decompositions
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.
scientific article; zbMATH DE number 4127414
scientific article