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The generalized bisymmetric solutions of the matrix equation A 1  X 1  B 1 + A 2  X 2  B 2 +  ⋯ + A l   X l   B l = C and its optimal approximation

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Abstract

For fixed generalized reflection matrix P, i.e. P T = P, P 2 = I, then matrix X is said to be generalized bisymmetric, if X = X T = PXP. In this paper, an iterative method is constructed to find the generalized bisymmetric solutions of the matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 + ⋯ + A l X l B l  = C where [X 1,X 2, ⋯ ,X l ] is real matrices group. By this iterative method, the solvability of the matrix equation can be judged automatically. When the matrix equation is consistent, for any initial generalized bisymmetric matrix group \(\left[X_1^{(0)},X_2^{(0)},\cdots,X_l^{(0)}\right]\), a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm generalized bisymmetric solution group can be obtained by choosing a special kind of initial generalized bisymmetric matrix group. In addition, the optimal approximation generalized bisymmetric solution group to a given generalized bisymmetric matrix group \(\left[\overline{X}_1,\overline{X}_2,\cdots,\overline{X}_l\right]\) in Frobenius norm can be obtained by finding the least norm generalized bisymmetric solution group of the new matrix equation \(A_1\widetilde{X}_1B_1+A_2\widetilde{X}_2B_2+\cdots+A_l\widetilde{X}_lB_l=\widetilde{C}\), where \(\widetilde{C}=C-A_1\overline{X}_1B_1-A_2\overline{X}_2B_2-\cdots-A_l\overline{X}_lB_l\). Given numerical examples show that the algorithm is efficient.

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Authors and Affiliations

  1. School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, 411201, People’s Republic of China

    Zhuo-hua Peng & Jin-wang Liu

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  1. Zhuo-hua Peng
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  2. Jin-wang Liu
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Correspondence to Zhuo-hua Peng.

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Research supported by: (1) the National Natural Science Foundation of China (10571047) and (10771058), (2) Natural Science Foundation of Hunan Province (06JJ2053), (3) Scientific Research Fund of Hunan Provincial Education Department(06A017).

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Peng, Zh., Liu, Jw. The generalized bisymmetric solutions of the matrix equation A 1  X 1  B 1 + A 2  X 2  B 2 +  ⋯ + A l   X l   B l = C and its optimal approximation. Numer Algor 50, 127–144 (2009). https://doi.org/10.1007/s11075-008-9220-9

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  • DOI: https://doi.org/10.1007/s11075-008-9220-9

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