Inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation

In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors { x i} i=1 m in C n and a set of complex numbers { λ i } i=1 m , find a centrosymmetric or centroskew matrix C in R n×n such that { x i} i=1 m and { λ i } i=1 m 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 R 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.