Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix

Fan-Liang Li

doi:10.1515/math-2020-0020

Abstract

Abstract Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.

Highlights

Throughout this article we use some notations as follows
Our results extend and unify many results for left and right inverse eigenpairs problem, the inverse problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint, which is the first motivation of this work
We derive the suitable supposition for Problem I (X, Y, Λ, Γ are given by (2.5)), which is another motivation of this work

Summary

Introduction

Throughout this article we use some notations as follows. Let Cn×m be the set of all n × m complex matrices, Rn×m be the set of all n × m real matrices, Cn = Cn×1, Rn = Rn×1, R denote the set of all real numbers, ORn×n denote the set of all n × n orthogonal matrices, R(A), AT, r(A), tr(A) and A+ be the column space, the transpose, rank, trace and the Moore–Penrose generalized inverse of a matrix A, respectively. The left and right inverse eigenpairs problem is as follows: given partial left and right eigenpairs (eigenvalue and corresponding eigenvector) (γj, yj), j = 1,...,l; (λi, xi), i = 1,...,h, and a special n × m matrix set S, to find A ∈ S such that. This recursive inverse eigenvalue problem is a special case of the left and right inverse eigenvalue problem with the leading principal submatrix constraint. We will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix, which has not been discussed.

Solvability conditions of Problem I

An expression of the solution of Problem II

Conclusion