On a class of matrices generated by certain generalized permutation matrices and applications

ABSTRACT

In this paper, we study a particular class of matrices generated by generalized permutation matrices corresponding to a subgroup of some permutation group. As applications, we first present a technique from which we can get closed formulas for the roots of many families of polynomial equations with degree between 5 and 10, inclusive. Then, we describe a tool that shows how to find solutions to Fermat's last theorem and Beal's conjecture over the square integer matrices of any dimension. Finally, simple generalizations of some of the concepts in number theory to integer square matrices are presented.

KEYWORDS:

• Generalized permutation matrices
• eigenvalues
• eigenvectors
• roots of polynomials
• non-negative matrices
• inverse eigenvalue problem
• matrix number theory

2010 AMS SUBJECT CLASSIFICATIONS:

• 15A18
• 15A51
• 15A24
• 15A29
• 12D10

Acknowledgements

The authors sincerely thank the reviewer and the editor for their valuable comments and helpful suggestions on an early version of this manuscript, which led to a substantial improvement on the presentation and contents of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported from LIU. The second author acknowledges financial support from the Lebanese University (Université Libanaise) research grants program.