The paper is concerned with estimating the spectral radius $ρ(A)$ of a square real matrix $A$. In the following, $\|\cdot\|_H$ denotes the operator norm induced by the weighted $2$-norm with a symmetric positive definite weight matrix $H$. Based on the classical result that the Stein equation $A^ H A - H = -W$ has a unique symmetric positive definite solution $H$ for every symmetric positive definite right-hand side $W$ if and only if $ρ(A)<1$, the following characterization of $ρ(A)$ is derived: If $ρ(A)<1$ then there is a symmetric positive definite matrix $H$ such that $$ A^T H A - H = -I,\quad \|A\|_H = \sqrt{1-1/λ_{\max}(H)}, $$ where $λ_{\max}$ denotes the maximal eigenvalue of a symmetric matrix. This characterization is turned into an iteration, which produces a sequence $H_0,H_1,H_2,\ldots$ such that $\|A\|_{H_k}$ decreases monotonically and converges to $ρ(A)$. Hence, in contrast to more established methods, such as the power method, each iterate provides an upper bound on the spectral radius. However, each iteration requires the solution of a Stein equation, which is usually significantly more expensive than a matrix-vector multiplication.
Reviewer: Daniel Kressner (Lausanne)