An estimation of the spectral radius of a product of block matrices

Let C(r)=[Cij], r=1,2,…,R, be block m×m matrices where Cij(r) are nonnegative Ni×Nj matrices for i,j=1,2,…,m. Let • be a consistent matrix norm. Denote for each r by B(r)=[ Cij(r) ] an m×m matrix. The relation of the spectral radii ρ(∏r=1RC(r)) and ρ(∏r=1RB(r)) is studied in this paper. It is shown with two proofs that Asshown in one of the proofs, ρ(∏r=1RB(r)) can be reduced so that it gives a better estimation of ρ(∏r=1RC(r)).