Continuity of the Perron root

Abstract

That the Perron root of a square nonnegative matrix varies continuously with the entries in is a corollary of theorems regarding continuity of eigenvalues or roots of polynomial equations, the proofs of which necessarily involve complex numbers. But since continuity of the Perron root is a question that is entirely in the field of real numbers, it seems reasonable that there should exist a development involving only real analysis. This article presents a simple and completely self-contained development that depends only on real numbers and first principles.

Keywords:

• Perron root
• Perron–Frobenius theory
• nonnegative matrices

AMS Subject Classifications:

• 1502
• 15A18
• 15B48

Acknowledgements

The author wishes to thank the referee for providing suggestions and corrections that enhanced the exposition. The referee is also responsible for example (66 ), and for pointing out the work in [4]. In addition, thanks are extended to Stephen Campbell for suggesting the simple explanation of why the convergence of (55 ) is not uniform.