Inverse Eigenproblems and Associated Approximation Problems for Matrices with Generalized Symmetry or Skew Symmetry

Let R∈Cn×n be a nontrivial involution; i.e., R=R−1≠±I. We say that A∈Cn×n is R-symmetric (R-skew symmetric) if RAR=A (RAR=−A). Let S be one of the following subsets of Cn×n: (i) R-symmetric matrices; (ii) Hermitian R-symmetric matrices; (iii) R-skew symmetric matrices; (iv) Hermitian R-skew symmetric matrices. Let Z∈Cn×m with rank(Z)=m and Λ=diag(λ1,…,λm). The inverse eigenproblem consists of finding (Z,Λ) such that the set S(Z,Λ)={A∈S|AZ=ZΛ} is nonempty, and to find the general form of A∈S(Z,Λ). In all cases we use the special spectral properties of S to essentially characterize the set of admissible pairs (Z,Λ), and the special structure of the members of S to obtain the general solution of the inverse eigenproblem. Given an arbitrary B∈S, the approximation problem consists of finding the unique matrix AB∈S(Λ,Z) that best approximates B in the Frobenius norm. It is not necessary to assume that R=R* in connection with the inverse eigenproblem for R-symmetric or R-skew symmetric matrices. However, we impose this additional assumption in connection with the inverse eigenproblem for Hermitian R-symmetric or R-skew symmetric matrices, and in connection with the approximation problem for (i)–(iv).