On the limit products of a family of matrices

Nicola Guglielmi, Marino Zennaro

Dipartimento di Matematica Pura e Applicata, Università dell'Aquila, 67100 L'Aquila, Dipartimento di Scienze Matematiche, Università di Trieste, 34100 Trieste (Italy)

guglielm@univaq.it, zennaro@univ.trieste.it

http://mathsun1.univ.trieste.it/~guglielm/guglielmi_eng.html, http://www.dsm.univ.trieste.it/~zennaro/index.html

Contributed talk

A linear difference equation with variable coefficients can be naturally studied in its companion matrix form. Doing this we are driven to the analysis of a family of $n\!\times \!n$ square matrices, ${\cal F}=\{A^{(i)}\}_{i\in {\cal I}}$ (where ${\cal I}$ is a set of indexes). Typically ${\cal F}$ depends on certain parameters (e.g. the (variable) stepsize of integration adopted by a numerical method for differential equations). Analyzing the stability properties of the solution is then restated as the problem of determining the unconditional asymptotic convergence of all possible products of matrices of the family. In this talk we direct attention at bounded families of complex $n\!\times \!n$-matrices. In order to study their asymptotic behaviour, we recall from [2] the concept of limit spectrum-maximizing product and show that non-defective families always admit such limit products. Then we consider defective families. In [2] we proved that, for finite families of $2\!\times \!2$-matrices, defectivity is equivalent to the existence of defective such limit products. This result led us to conjecture the validity of this property also for higher dimensions $n \ge 3$. Here, instead, by making use of the results obtained by Bousch and Mairesse [1] that disproved the wellknown Finiteness Conjecture, we find some counterexamples to our conjecture in [2] for all $n \ge 3$.