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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