The left and right inverse eigenvalue problems of generalized reflexive and anti-reflexive matrices

Let n × n complex matrices R and S be nontrivial generalized reflection matrices, i.e.,  R ∗ = R = R − 1 ≠ ± I n , S ∗ = S = S − 1 ≠ ± I n . A complex matrix A with order n is said to be a generalized reflexive (or anti-reflexive ) matrix, if R A S = A (or R A S = − A ). In this paper, the solvability conditions of the left and right inverse eigenvalue problems for generalized reflexive and anti-reflexive matrices are derived, and the general solutions are also given. In addition, the associated approximation solutions in the solution sets of the above problems are provided. The results in present paper extend some recent conclusions.