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

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