Construction of Optimal Norms for Semi-Groups of Matrices

The notion of spectral radius of a set of matrices is a natural extension of spectral radius of a single matrix. The Finiteness Conjecture (FC) claims that among the infinite products made from the elements of a given finite set of matrices, there is a certain periodic product, made from the repetition of a finite product (the optimal product), whose rate of growth is maximal. FC has been disproved. In this paper it is conjectured that FC is almost always true, and an algorithm is presented to verify the optimality of a given product. The algorithm uses optimal norms, as a special subset of extremal extremal norms. The algorithm has successfully calculated the spectral radius of the pair of matrices associated with compactly supported multi-resolution analyses and wavelets.