Inverse eigenproblem for R-symmetric matrices and their approximation

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Abstract

Let RCn×n be a nontrivial involution, i.e., R=R1±In. We say that GCn×n is R-symmetric if RGR=G. The set of all n×nR-symmetric matrices is denoted by GSCn×n. In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors {xi}i=1m in Cn and a set of complex numbers {λi}i=1m, find a matrix AGSCn×n such that {xi}i=1m and {λi}i=1m are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix Ã, find AˆSE such that ÃAˆ=minASEÃA, where SE is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution Aˆ by means of the canonical correlation decomposition.

MSC

65F18
15A24
15A57

Keywords

Inverse eigenproblem
R-symmetric matrix
Canonical correlation decomposition (CCD)
Best approximation

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