Inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation

摘要

In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors {X{sub}i}{sub}i{sup}m in C{sup}n and a set of complex numbers {λ{sub}i}{sub}i{sup}m, find a centrosymmetric or centroskew matrix C in R{sup}(n × n) such that {X{sub}i}{sub}i{sup}m and {λ{sub}i}{sub}i{sup}m are the eigenvectors and eigenvalues of C, respectively, We then consider the best approximation problem for the lEPs that are solvable. More precisely, given an arbitrary matrix B in R{sup}(n × 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.