Inverse Eigenvalue Problem and Least-Squares Problem for Skew-Hermitian {P,K + 1}-Reflexive Matrices

Abstract

This paper involves related inverse eigenvalue problem and least-squares problem of skew-Hermitian {P,k + 1}-reflexive(antireflexive) matrices and their optimal approximation problems. The above problems are studied by converting them into two simpler cases: k = 1 and k = 2. Firstly, with some special properties of skew-Hermitian {P,k + 1}-reflexive(antireflexive) matrices, the necessary and sufficient conditions for the solvability and the general solution are presented, and the solution of corresponding optimal approximation problems also given, respectively. Then, we give the least-squares solution of $\text{AX}=B$ satisfying the special condition by the singular value decomposition. Finally, we give an algorithm and an example to illustrate our results.