Class07

In this talk, I will survey several aspects of an apparently simple (but so far unsolved) question: under what conditions on the matrices A and B do all possible infinite products of the type ABBABAAAB… converge to zero? This question has implications for a number of practical problems for hybrid systems, wavelets, codes and sensor networks which I will describe in the talk. The question can be formalized by introducing the concept of joint spectral radius that measures the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from a set. The joint spectral radius appears in a number of application contexts but is notoriously difficult to compute and to approximate. I will prove that the problem of determining if the set all possible products of two matrices is bounded, is algorithmically undecidable and will exhibit occurrences of Sturmian sequences in the analysis of optimal periodic products. I will then conclude the talk with a recent efficient approximation algorithm for the joint spectral radius. The algorithm approximates the joint spectral with arbitrary high accuracy and is polynomial in the size of the matrices once the desired accuracy is fixed.

Vincent D. Blondel (born on April 28) is visiting professor and Fulbright Scholar at the Massachusetts Institute of Technology and head of the Department of Mathematical Engineering of the Universite catholique de Louvain in Belgium. He received a Master Degree in Pure Mathematics from Imperial College (London, UK) and a PhD in Applied Mathematics from the Universite Catholique de Louvain in 1992. He was the Gian Gustafsson Fellow at the Royal Institute of Technology (Stockholm, Sweden) in 1992-1993 and was a research fellow a the French National Research Center in Control and Computer Science (INRIA, Rocquencourt-Paris) in 1993-1994. He has held research and teaching positions at the University of California at Berkeley, the Santa Fe Institute, the Ecole Normale Supieure in Lyon and the University of Paris VII. He has been a recipient of a Grant from the Trustees of the Mathematics Institute of Oxford University, the Prize Agathon De Potter of the Belgian Royal Academy of Science, the Prize Paul Dubois of the Montefiore Institute, and was awarded the triennal SIAM prize on control and systems theory.

Host: Prof. Paschalidis
Student Host: Kirk Wesselowski