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Inverse problems for symmetric matrices with a submatrix constraint. (English) Zbl 1134.65034

Suppose \(X\in \mathbb R^{n\times p}\) is a full column rank matrix and \(B\in \mathbb R^{p\times p}\), \(A_0\in \mathbb R^{r\times r}\) are symmetric matrices. The authors consider the following problems:
(a) Find a symmetric matrix \(A\in \mathbb R^{n\times n}\) such that \(X^\perp A X=B\) and \(A[1, r] = A_0\);
(b) Find a symmetric matrix \(A\in \mathbb R^{n\times n}\) such that \(\| X^\perp A X-B\| =\min\) and \(A[1, r] = A_0\);
(c) Given a symmetric matrix \(\widetilde A\in \mathbb R^{n\times n}\), find \(\widehat A\in S_E\) such tht \(\| \widehat A-\widetilde A\| =\inf_{A\in S_E} \| A-\widetilde A\| \).
Here \(A[1, r]\) denotes the leading principal \(r\times r\) submatrix of \(A\) and \(S_E\) is the set of solutions for the problem (a). The main tools used to discuss the above problems are the generalized singular value decomposition (GSVD) and the canonical correlation decomposition (CCD) of a matrix pair. Existence of a solution for (a) is characterized by requiring the matrix \(A_0\) to satisfy certain relation in terms of \(B\) and \(X\) using the GSVD of the matrix pair \([X_1^\perp, X_2^\perp]\) where \(X=[X_1^\perp, X_2^\perp]^\perp\) with \(X\in \mathbb R^{r\times p}\), \(X_2\in \mathbb R^{(n-r)\times p}\). Also given are expressions for the solutions to the problems (b) and (c). An algorithm is given for (a) and (b) and the procedure is illustrated by a numerical example.

MSC:

65F30 Other matrix algorithms (MSC2010)
15A29 Inverse problems in linear algebra
65F20 Numerical solutions to overdetermined systems, pseudoinverses
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References:

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