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 {xi}i=1 in Cn and a set of complex numbers {λi}i=1, find a centrosymmetric or centroskew matrix C in Rn×n such that {xi}i=1 and {λi}i=1 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.