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The bisymmetric solutions of the matrix equation \(A_{1}X_{1}B_{1}+A_{2}X_{2}B_{2}+\cdots+A_{l}X_{l}B_{l}=C\) and its optimal approximation. (English) Zbl 1123.15009

The authors propose an iterative method for finding the bisymmetric solutions of the matrix equation \(A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l=C~(*)\) where \([X_1,\dots ,X_l]\) is a real matrix group. (A square matrix is bisymmetric if it is symmetric w.r.t. both the diagonal and the antidiagonal.)
The method gives also the answer to the question whether equation \((*)\) is consistent or not. When it is, then for any initial bisymmetric matrix group \([X_1^{(0)},\ldots ,X_l^{(0)}]\) a solution is obtained within finitely many iteration steps in the absence of roundoff errors, and the choice of a special initial matrix group allows one to find the least norm bisymmetric solution group (LNBSG).
The optimal approximation bisymmetric solution group to a given group \([\bar{X}_1,\dots ,\bar{X}_l]\) in the Frobenius norm is obtained by finding the LNBSG of equation \((*)\) with right-hand side \(C-A_1\bar{X}_1B_1-\cdots -A_l\bar{X}_lB_l\).

MSC:

15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
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References:

[1] Peng, Zhen-yun, The inverse problem of bisymmetric matrices, Numer. Linear Algebra Appl., 11, 59-73 (2004) · Zbl 1164.15322
[2] Peng, Zhen-yun, The solutions of matrix \(AXC + BYD = E\) and its optimal approximation, Math. Theory Appl., 22, 2, 99-103 (2002) · Zbl 1504.15053
[3] Shim, S.-Y.; Chen, Y., Least squares solution of matrix equation \(AXB^* + CYD^* = E\), SIAM J. Matrix Anal. Appl., 3, 802-808 (2003) · Zbl 1037.65042
[4] Chu, K. E., Singular value and generalized value decompositions and the solution of linear matrix equations, Linear Algebra Appl., 87, 83-98 (1987) · Zbl 0612.15003
[5] Peng, Zhen-yun, The nearest bisymmetric solutions of linear matrix equations, J. Comput. Math., 22, 6, 873-880 (2004) · Zbl 1068.65057
[6] Dai, H., On the symmetric solutions of linear matrix equations, Linear Algebra Appl., 131, 1-7 (1990) · Zbl 0712.15009
[7] Baruch, M., Optimization procedure to correct stiffness and flexibility matrices using vibration tests, AIAA J., 16, 1208-1210 (1978) · Zbl 0395.73056
[8] Jeseph, K. T., Inverse eigenvalue problem in structural design, AIAA J., 30, 2890-2896 (1992) · Zbl 0825.73453
[9] Jiang, Z.; Lu, Q., Optimal application of a matrix under spectral restriction, Math. Numer. Sinica, 1, 47-52 (1988) · Zbl 0592.65023
[10] Xie, Dong-xiu; Zhang, L.; Hu, X. Y., The solvability conditions for the inverse problem of bisymmetric nonnegative definite matrices, J. Comput. Math., 6, 597-608 (2000) · Zbl 0966.15008
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