Peng, Zhen-yun; Hu, Xi-yan; Zhang, Lei The inverse problem of bisymmetric matrices with a submatrix constraint. (English) Zbl 1164.15322 Numer. Linear Algebra Appl. 11, No. 1, 59-73 (2004). 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. Cited in 36 Documents MSC: 15A29 Inverse problems in linear algebra 15B57 Hermitian, skew-Hermitian, and related matrices 65F30 Other matrix algorithms (MSC2010) Keywords:bisymmetric matrix PDFBibTeX XMLCite \textit{Z.-y. Peng} et al., Numer. Linear Algebra Appl. 11, No. 1, 59--73 (2004; Zbl 1164.15322) Full Text: DOI