On the asymptotic properties of a family of matrices

In this paper we consider bounded families F of complex n x n-matrices. Aft er introducing the concept of asymptotic order, we investigate how the norm of products of matrices behaves as the number of factors goes to infinity. In the case of defective families F, using the asymptotic order allows us to get a deeper knowledge of the asymptotic behaviour than just considering the so-called generalized spectral radius. With reference to the well-know n finiteness conjecture for finite families, we also introduce the concepts of spectru-maximizing product and limit spectrum-maximizing product, showi ng that, for finite families of 2 x 2-matrices, defectivity is equivalent t o the existence of defective such limit products. (C) 2001 Elsevier Science Inc. All rights reserved.