On bounding the eigenvalues of matrices with constant row-sums

ABSTRACT

For every $n\times n$ real matrix A having as an eigenvector e, the all-ones vector, an upper bound is obtained for the absolute value of its eigenvalues except, in some cases, the one associated with e. Comparisons are made between this bound and some existing bounds for stochastic matrices as well as other types of matrices such as Laplacian matrices and Randić matrices. Other examples and results are also provided. An interesting recent article ‘An eigenvalue localization theorem for stochastic matrices and its application to Randić matrices’, Linear Algebra Appl. 2016;505:85–96, by A. Banerjee and R. Mehatari, is useful for this present paper.

KEYWORDS:

• Stochastic matrix
• Laplacian matrix
• eigenvalue
• upper bound

Acknowledgements

The authors wish to thank Roger Horn for his very valuable assistance. The author Rachid Marsli would like to express sincere appreciation to Abduslam Mimouni from King Fahd University of Minerals and Petroleum.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by King Fahd University of Minerals and Petroleum [SR161014].