Inverse eigenproblem for RR-symmetric matrices and their approximation

Let R∈Cn×n be a nontrivial involution, i.e., R=R−1≠±InR=R−1≠±In. We say that G∈Cn×n is RR-symmetric if RGR=GRGR=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 A∈GSCn×n such that {xi}i=1m and {λi}i=1m are, respectively, the eigenvalues and eigenvectors of AA. We then consider the following approximation problem: Given an n×nn×n matrix Ã, find Aˆ∈SE such that ‖Ã−Aˆ‖=minA∈SE‖Ã−A‖, 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.