An efficient lower bound for the generalized spectral radius of a set of matrices

AbstractThe generalized spectral radius (GSR) is a fundamental concept in studying the regularity of compactly supported wavelets. Here we describe an efficient method for estimating a lower bound for the GSR. Let Mq be the set of all q × q matrices with complex entries. Suppose ∑ = {A0, …, Am−1} is a collection of m matrices in Mq. Let Ln be the set of all products of length n of the elements of ∑. Define n(∑) = maxA∈Ln [ϱ(A)]In, where ϱ(A) is the spectral radius of A. The generalized spectral radius of ∑ is then ϱ(∑) = lim supn → ∞ ϱn(∑). The standard method for estimating ϱ(∑), from below and at level n, is to calculate the spectral radii of all mn products in Ln and select the largest. Here we use three elementary theorems from linear algebra, combinatorics, and number theory to show that the same result can be obtained with no more than mnn matrix calculations