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An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation $AXB$=$C$. (English) Zbl 1068.65056
Let $\Bbb R^{m\times n}$ be the set of all $m\times n$ matrices, $S\Bbb R^n$ the set of all symmetric matrices in $\Bbb R^{n\times n}$. For $A\in \Bbb R^{m\times n}$, $\Vert A\Vert $ denotes the Frobenius norm. The authors consider the following two problems. Problem 1. Given $A\in \Bbb R^{m\times n}$, $B\in \Bbb R^{n\times p}$, $C\in \Bbb R^{m\times p}$, find $X\in S\Bbb R^{n}$ such that $AXB=C$. Problem 2. If Problem 1 is consistent, then denote its solutions by ${\cal S}_E$. For given $X_0\in \Bbb R^{n\times n}$, find $\hat{X}\in {\cal S}_E$ such that $$ \Vert \hat{X}-X_0\Vert = \min \{\Vert X-X_0\Vert :X\in {\cal S}_E \}. $$ The authors describe an iterative method that determines the solvability of Problem 1 automatically and in the case of solvability computes a solution in an a priori known finite number of steps. Furthermore, the solution to Problem 2 can be found by choosing a suitable initial iteration matrix. It can also be found as the least-norm solution to another equation $A\bar{X}B=\bar{C}$. The paper is carefully written with detailed and convincing proofs. It also contains a numerical example.

MSC:
65F30Other matrix algorithms
65F10Iterative methods for linear systems
15A24Matrix equations and identities
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References:
[1] Golub, G. H.; Van Loan, C. F.: Matrix computations. (1996) · Zbl 0865.65009
[2] Ben-Israel, A.; Greville, T. N. E.: Generalized inverses, theory and applications [M]. (1974) · Zbl 0305.15001
[3] Hua, Dai: On the symmetric solutions of linear matrix equations. Linear algebra appl 131, 1-7 (1990) · Zbl 0712.15009
[4] Chu, King-Wah Eric: Symmetric solutions of linear matrix equations by matrix decompositions. Linear algebra appl 119, 35-50 (1989) · Zbl 0688.15003
[5] Don, F. J. Henk: On the symmetric solution of a linear matrix equation. Linear algebra appl 93, 1-7 (1988)
[6] Magnus, J. R.: L-structured matrices and linear matrix equation. Linear and multilinear algebra appl 14, 67-88 (1983) · Zbl 0527.15006
[7] Morris, G. R.; Odell, P. L.: Common solutions for n matrix equations with applications. J. assoc. Comput. Mach 15, 272-274 (1968) · Zbl 0157.22602
[8] Mitra, S. K.: Common solutions to a pair of linear matrix equations A1XB1=C1, A2XB2=C2. Proc. Cambridge philos., soc 74, 213-216 (1973)
[9] C.R. Rao, Generalized inverse for matrices and its applications in mathematical statistics, Research Papers in Statistics, Festschrift for J. Neyman, John Wiley, New York, 1965
[10] Penrose, R.: A generalized inverse for matrices. Proc. Cambridge philos., soc 51, 406-413 (1955) · Zbl 0065.24603
[11] Bjerhammer, A.: Rectangular reciprocal matrices with special reference to geodetic calculations. Kung tekn. Hogsk. handl. Stockholm 45, 1-86 (1951)