An iterative algorithm for the least squares bisymmetric solutions of the matrix equations \(A_{1}XB_{1}=C_{1},A_{2}XB_{2}=C_{2}\). (English) Zbl 1190.65061

The authors propose an iterative algorithm for solving the minimum Frobenius norm residual problem \(\min \left[ \left( A_{1}XB_{1}-C_{1} \right)^2 + \left( A_{2}XB_{2}-C_{2} \right)^2 \right]\) over bisymmetric matrices. The algorithm acts on the associated normal equation of the initial one.


65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
15A24 Matrix equations and identities
Full Text: DOI


[1] Mitra, S. K., Common solutions to a pair of linear matrix equations \(A_1 X B_1 = C_1, A_2 X B_2 = C_2\), Proc. Cambridge Philos. Soc., 74, 213-216 (1973)
[2] Mitra, S. K., A pair of simultaneous linear matrix equations and a matrix programming problem, Linear Algebra Appl., 131, 97-123 (1990)
[3] Shinozaki, N.; Sibuya, M., Consistency of a pair of matrix equations with an application, Kieo Eng. Rep., 27, 141-146 (1974)
[5] Navarra, A.; Odell, P. L.; Young, D. M., A representation of the general common solution to the matrix equations \(A_1 X B_1 = C_1, A_2 X B_2 = C_2\) with applications, Comput. Math. Appl., 41, 929-935 (2001) · Zbl 0983.15016
[6] Yuan, Y. X., Least squares solutions of matrix equation \(A X B = E, C X D = F\), J. East China Shipbuilding Inst., 18, 3, 29-31 (2004)
[7] Deng, Y. B.; Bai, Z. Z.; Gao, Y. H., Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations, Numer. Linear Algebra Appl., 13, 801-823 (2006) · Zbl 1174.65382
[8] Peng, Y. X.; Hu, X. Y.; Zhang, L., An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations, Appl. Math. Comput., 183, 1127-1137 (2006) · Zbl 1134.65032
[9] Sheng, X. P.; Chen, G. L., A finite iterative method for solving a pair of linear matrix equations \((A X B, C X D) = (E, F)\), Appl. Math. Comput., 189, 1350-1358 (2007) · Zbl 1133.65026
[10] Meng, T., Experimental design and decision support, (Leondes, Expert System, the Technology of Knowledge Management and Decision Making for the 21st Century. Vol. 1 (2001), Academic Press)
[11] Higham, N. J., Computing a nearest symmetric positive semidefinite matrix, Linear Algebra Appl., 103, 103-118 (1988) · Zbl 0649.65026
[12] Jiang, Z.; Lu, Q., On optimal approximation of a matrix under a spectral restriction, Math. Numer. Sin., 8, 47-52 (1986) · Zbl 0592.65023
[13] Peng, Z. Y.; Hu, X. Y.; Zhang, L., The inverse problem of bisymmetric matrices, Numer. Linear Algebra Appl., 1, 59-73 (2004) · Zbl 1164.15322
[14] Antoniou, A.; Lu, W. S., Practical Optimization: Algorithm and Engineering Applications (2007), Springer: Springer New York, (Chapter 2)
[15] Peng, Z. H.; Hu, X. Y.; Zhang, L., The bisymmetric solutions of the matrix equation \(A_1 X B_1 + A_2 X B_2 + \cdots + A_l X B_l = C\) and its optimal approximation, Linear Algebra Appl., 426, 583-595 (2007)
[16] Ben-Israel, A.; Greville, T. N.E., Generalized Inverse: Theory and Applications (2002), Wiley: Wiley New York
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