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An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations $A_{1}XB_{1} = C_{1}, A_{2}XB_{2} = C_{2}$. (English) Zbl 1134.65032
The symmetric solutions of the systems of matrix equations $A_1XB_1=C_1$, $A_2XB_2=C_2$ can not be easily obtained by applying matrix decompositions. The authors are proposing an iterative method to solve systems of matrix equations, where when the system of matrix equations is consistent, and its solution can be obtained within finite iterative steps, and its least-norm solution can be obtained by choosing a special kind of initial iterative matrix. Additionally, its optimal approximation solution to a given matrix can be derived by finding the least-norm symmetric solution of a new system of matrix equations $A_1\hat XB_1=\hat C_1$, $A_2\hat XB_2=\hat C_2$. Finally, the author demonstrates the applicability of the proposed method on systems of matrix equations.

MSC:
65F30Other matrix algorithms
15A24Matrix equations and identities
65F10Iterative methods for linear systems
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References:
[1] Mitra, S. K.: The matrix equations AX=C, XB=D. Linear algebra appl. 59, 171-181 (1984) · Zbl 0543.15011
[2] Mitra, S. K.: A pair of simultaneous linear matrix equations A1XB1=C1, and A2XB2=C2 and a matrix programming problem. Linear algebra appl. 131, 107-123 (1990)
[3] Navarra, A.; Odell, P. L.; Young, D. M.: A representation of the general common solution to the matrix equations A1XB1=C1, and A2XB2=C2 with applications. Comput. math. Appl. 41, 929-935 (2001) · Zbl 0983.15016
[4] Bhimasankaram, P.: Common solutions to the linear matrix equations AX=B, CX=D, and EXF=G. Sankhya ser. A 38, 404-409 (1976) · Zbl 0411.15008
[5] Van Der Woude, J. W.: On the existence of a common solution X to the matrix equations aixbj=Cij, (i,j)$\in \gamma $. Linear algebra appl. 375, 135-145 (2003) · Zbl 1037.15014
[6] Hestenes, M. R.; Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. res. Nat. bur. Stand. 49, 409-436 (1952) · Zbl 0048.09901
[7] Hestenes, M. R.: Conjugate direction methods in optimization. (1980) · Zbl 0439.49001
[8] Reid, J. K.: On the method of conjugate gradients for the solution of large sparse systems of linear equations. Large sparse sets of linear equations, 231-254 (1971)
[9] Young, D. M.; Jea, K. C.: Generalized conjugate gradient acceleration of nonsymmetrizable iterative methods. Linear algebra appl. 34, 159-194 (1980) · Zbl 0463.65025
[10] Axelsson, O.: Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations. Linear algebra appl. 29, 1-16 (1980) · Zbl 0439.65020
[11] Golub, Gene H.; Van Loan, Charles F.: Matrix computations. (1996) · Zbl 0865.65009
[12] Peng, Ya-Xin; Hu, Xi-Yan; Chang, Lei: An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C. Appl. math. Comput. 160, 763-777 (2005) · Zbl 1068.65056