Inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation

Authors:
Zheng-Jian Bai;Raymond H. Chan
Affiliations:
Department of Mathematics, Chinese University of Hong Kong, Shatin, NT, Hong Kong, China;Department of Mathematics, Chinese University of Hong Kong, Shatin, NT, Hong Kong, China
Venue:
Theoretical Computer Science - Algebraic and numerical algorithm
Year:
2004

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Abstract

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 centrosymmetric or centroskew matrix C in Rn × n such that {Xi}i-1m and {λi}i-1m are the eigenvectors and eigenvalues of C, respectively. We then consider the best approximation problem for the IEPs that are solvable. More precisely, given an arbitrary matrix B in Rn × n, we find the matrix C which is the solution to the IEP and is closest to B in the Frobenius norm. We show that the best approximation is unique and derive an expression for it.