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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.

MSC:
65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
15A24 Matrix equations and identities
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[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)
[4] J.W. van der Woude, Freeback decoupling and stabilization for linear systems with multiple exogenous variables, Ph.D. Thesis, 1987, pp. 85-199
[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, ()
[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 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 New York
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