Inverse eigenproblem for -symmetric matrices and their approximation

Let R ∈ C n × n be a nontrivial involution, i.e., R = R − 1 ≠ ± I n . We say that G ∈ C n × n is R -symmetric if R G R = G . The set of all n × n R -symmetric matrices is denoted by GSC n × n . 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 matrix A ∈ GSC n × n such that { x i } i = 1 m and { λ i } i = 1 m are, respectively, the eigenvalues and eigenvectors of A . We then consider the following approximation problem: Given an n × n matrix A ̃ , find A ˆ ∈ S E such that ‖ A ̃ − A ˆ ‖ = min A ∈ S E ‖ A ̃ − A ‖ , where S E is the solution set of IEP and ‖ ⋅ ‖ is the Frobenius norm. We provide an explicit formula for the best approximation solution A ˆ by means of the canonical correlation decomposition.