Inverse eigenvalue problems for bisymmetric matrices under a central principal submatrix constraint

Abstract

This article considers an inverse eigenvalue problem for bisymmetric matrices under a central principal submatrix constraint and the corresponding optimal approximation problem. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we study a special form for the matrix of independent eigenvectors for a bisymmetric matrix. Based on these, we give some necessary and sufficient conditions for the solvability of the inverse eigenvalue problem, and we derive an expression for its general solution. Finally, we obtain an expression for the solution to the corresponding optimal approximation problem.

Keywords:

• bisymmetric matrix
• central principal submatrix
• inverse eigenvalue problem
• optimal approximation problem

AMS Subject Classifications:

• 15A57
• 15A18
• 65F35

Acknowledgements

The authors would like to thank Prof. Volker Mehrmann, Prof. Kerstin Ullrich and the referees very much for their valuable suggestions and comments, which have greatly improved this article. This research was supported by the National Natural Science Foundation of China (Grant No. 10571047) and Doctorate Foundation of the Ministry of Education of China (Grant No. 20060532014).