Perturbing eigenvalues of nonnegative centrosymmetric matrices

Abstract

An $n\times n$ matrix C is said to be centrosymmetric if it satisfies the relation JCJ = C, where J is the $n\times n$ counteridentity matrix. Centrosymmetric matrices have a rich eigenstructure that has been studied extensively in the literature. Many results for centrosymmetric matrices have been generalized to wider classes of matrices that arise in a wide variety of disciplines. In this paper, we obtain interesting spectral properties for nonnegative centrosymmetric matrices. We show how to change one single eigenvalue, two or three eigenvalues of an $n\times n$ nonnegative centrosymmetric matrix without changing any of the remaining eigenvalues, the nonnegativity, or the centrosymmetric structure. Moreover, our results allow partially answer some known questions given by [Guo W. Eigenvalues of nonnegative matrices. Linear Algebra Appl. 266;1997:261–270] and by [Guo S, Guo W. Perturbing non-real eigenvalues of non-negative real matrices. Linear Algebra Appl. 426;2007:199–203]. Our proofs generate algorithmic procedures that allow one to compute a solution matrix.

Keywords:

• Nonnegative matrices
• inverse eigenvalue problem
• centrosymmetric matrices
• spectral perturbation
• Guo's results

AMS classifications:

• 15A18
• 15A29
• 15A42

Acknowledgements

The authors express their thanks to the referee for the valuable comments which led to an improve version of the paper. The proof of Theorem 3.3 is due to the referee. The original proof by the authors was longer.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of A. I. Julio was supported by Universidad Católica del Norte [VRIDT 036-2020, NÚCLEO 6 UCN VRIDT083-2020, Chile]. The research of Y. R. Linares was supported by ANID-Subdirección de Capital Humano/Doctorado Nacional/ [2021-21210056, Chile].