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 r=1 C(r)) and ρ( ∏R r=1 B(r)) is studied in this paper. It is shown with two proofs that ρ   R ∏ r=1 C(r)   ρ   R ∏ r=1 B(r)   . As shown in one of the proofs, ρ( ∏R r=1 B(r)) can be reduced so that it gives a better estimation of ρ( ∏R r=1 C(r)). © 2003 Elsevier Inc. All rights reserved.