Constrained inverse eigenproblem and associated approximation problem for anti-Hermitian R-symmetric matrices

Let R ∈ Cn×n be a nontrivial involution, i.e., R2 = I and R ≠ ±I. A ∈ Cn×n is called anti-Hermitian R-symmetric if A∗ = −A and RAR = A. The presentation and some properties for an arbitrary anti-Hermitian R-symmetric matrix with R∗ = R and the relations between the eigenproblem for A and the corresponding eigenproblems for anti-Hermitian matrices are given. Then the solutions of Constrained Inverse Eigenproblem and Approximation Problem are essentially decomposed into the same kind subproblems for anti-Hermitian matrices in complex field with smaller dimensions. The explicit solutions for the later subproblems are arrived. The corresponding problems which are the formulations of Constrained Inverse Eigenproblem and Approximation Problem in complex field was first given, then the solutions of Constrained Inverse Eigenproblem and Approximation Problem with R∗ = R are derived.