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

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.