An Extension of the Rayleigh Quotient to the Spectral Radius of Asymmetric Nonnegative Matrices

TL;DR

This paper extends the Rayleigh quotient to characterize the spectral radius of asymmetric nonnegative matrices using a diagonal similarity scaling, providing a new variational formula and comparing it to existing ones.

Contribution

It introduces a novel variational formula for the spectral radius of nonsymmetric nonnegative matrices using diagonal similarity scaling.

Findings

Derives a new variational characterization of the spectral radius.

Shows how the formula generalizes the classical Rayleigh quotient.

Compares the new formula with existing variational characterizations.

Abstract

The Rayleigh quotient, which provides the classical variational characterization of the spectral radius of Hermitian matrices, can be extended to nonsymmetric nonnegative irreducible matrices, ${\bf A}$, by the inclusion of a diagonal similarity scaling, to yield the variational formula $r({\bf A}) = \sup_{{\bf x} > {\bf 0}} \inf_{{\bf y} > {\bf 0}} {\bf x}^\top {\bf D_y A D_y}^{-1} {\bf x}/({\bf x}^\top {\bf x})$, where ${\bf D_y}$ is the diagonal matrix of the vector ${\bf y}$. Comparison is made to other variational formulae for the spectral radius.