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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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##### References:
  Bartels, R.H.; Stewart, G.W., Solution of the equation AX + XB = C, Comm. ACM, 15, 820-826, (1972) · Zbl 1372.65121  Chu, K.-W.E., Singular value and generalized singular value decompositions and the solution of linear matrix equations, Linear algebra appl., 88/8, 83-98, (1987) · Zbl 0612.15003  Chu, K.-W.E., The solution of the matrix equations AXB − CXD = E and (YA − DZ, YYC − BZ) = (E, F),, Linear algebra appl., 93, 93-105, (1987) · Zbl 0631.15006  Davis, C.; Kahan, W.M.; Weinberger, H.F., Norm preserving dilations and their applications to optimal error bounds, SIAM J. numer. anal., 19, 445-469, (1982) · Zbl 0491.47003  Don, F.J.H., On the symmetric solutions of a linear matrix equation, Linear algebra appl., 93, 1-7, (1987) · Zbl 0622.15001  F. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York. · Zbl 0085.01001  Golub, G.H.; Nash, S.; Van Loan, C., A Hessenberg-Schur method for the matrix problem AX + XB = C, IEEE trans. automat. control, AC-24, 909-913, (1979) · Zbl 0421.65022  Golub, G.H.; Van Loan, C.F., Matrix computations, (1983), Johns Hopkins U.P Baltimore · Zbl 0559.65011  Hodges, J.H., Some matrix equations over a finite field, Ann. mat. pura appl., 44, 4, 245-250, (1957) · Zbl 0082.02102  Lancaster, P., Explicit solutions of linear matrix equations, SIAM rev., 12, 544-566, (1970) · Zbl 0209.06502  Magnus, J.R., L-structured matrices and linear matrix equations, Linear and multilinear algebra, 14, 67-88, (1983) · Zbl 0527.15006  Magnus, J.R.; Neudecker, H., The commutation matrix: some properties and applications, Ann. statist., 7, 381-394, (1979) · Zbl 0414.62040  Magnus, J.R.; Neudecker, H., The elimination matrix: some lemmas and applications, SIAM J. algebraic discrete methods, 1, 422-449, (1980) · Zbl 0497.15014  Nashed, M.Z., Generalized inverses and applications, (1976), Academic New York · Zbl 0346.15001  Paige, C.C.; Saunders, M.A., Towards a generalized singular value decomposition, SIAM J. numer. anal., 18, 398-405, (1981) · Zbl 0471.65018  Stewart, G.W., Computing the CS-decomposition of a partitioned orthogonal matrix, Numer. math., 40, 297-306, (1982) · Zbl 0516.65016  Vetter, W.J., Vector structures and solutions of linear matrix equations, Linear algebra appl., 10, 181-188, (1975) · Zbl 0307.15003
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