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The inverse problem of bisymmetric matrices with a submatrix constraint. (English) Zbl 1164.15322

Summary: An \(n\times n\) real matrix \(A\) is called a bisymmetric matrix if \(A=A^{\text T}\) and \(A=S_nAS_n\), where \(S_n\) is an \(n\times n\) reverse unit matrix. This paper is mainly concerned with solving the following two problems:
(I) Given \(n\times m\) real matrices \(X\) and \(B\), and an \(r\times r\) real symmetric matrix \(A_0\), find an \(n\times n\) bisymmetric matrix \(A\) such that \[ AX=B, \quad A_0=A([1:r]) \] where \(A([1:r])\) is an \(r\times r\) leading principal submatrix of the matrix A.
(II) Given an \(n\times n\) real matrix \(A^{\ast }\), find an \(n\times n\) matrix \(\hat A\) in \(S_E\) such that \[ \| A^{\ast }-\hat {A}\| = \min _{A\in S_E}\| A^{\ast }-A\| \] where \(\| \cdot \| \) is the Frobenius norm, and \(S_E\) is the solution set of Problem I.
Necessary and sufficient conditions for the existence of and the expressions for the general solutions of (I) are given. The explicit solution, a numerical algorithm and a numerical example for (II) are provided.

MSC:

15A29 Inverse problems in linear algebra
15B57 Hermitian, skew-Hermitian, and related matrices
65F30 Other matrix algorithms (MSC2010)
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