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The centrosymmetric solutions of matrix equation \(A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l=C\) and its optimal approximation. (Chinese. English summary) Zbl 1199.65147

Summary: A matrix \(X=(x_{ij})\in R^{n\times n}\) is said to be centrosymmetric if \(x_{ij}=x_{n+1-i, n+1-j}(i, j=1, 2,\cdots, n)\). In this paper, an iterative method is constructed for finding the centrosymmetric solutions of matrix equation \(A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l=C\), where \([X_1, X_2, \cdots, X_l]\) is a real matrix group. By this iterative method, the solvability of the matrix equation can be judged automatically. When the matrix equation is consistent, for any initial centrosymmetric matrix group \([X_1^{(0)}, X_2^{(0)}, \cdots, X_l^{(0)}],\) a centrosymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm centrosymmetric solution group can be obtained by choosing a special kind of initial centrosymmetric matrix group. In addition, the optimal approximation centrosymmetric solution group to a given centrosymmetric matrix group \([\overline{X}_1, \overline{X}_2, \cdots, \overline{X}_l]\) in Frobenius norm can be obtained by finding the least norm centrosymmetric solution group of 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 \(\widetilde{C}=C-A_1\overline{X}_1B_1-A_2\overline{X}_2B_2-\cdots-A_l\overline{X}_lB_l\). Numerical examples show that the iterative method is efficient.

MSC:

65F30 Other matrix algorithms (MSC2010)
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F10 Iterative numerical methods for linear systems
15A24 Matrix equations and identities
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