On the asymptotic properties of a family of matrices

AbstractIn this paper we consider bounded families F of complex n×n-matrices. After 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-known finiteness conjecture for finite families, we also introduce the concepts of spectrum-maximizing product and limit spectrum-maximizing product, showing that, for finite families of 2×2-matrices, defectivity is equivalent to the existence of defective such limit products