A criterion of simultaneously symmetrization and spectral finiteness for a finite set of real 2-by-2 matrices

For K≥ 1, let there be given an arbitrary finite set A consisting of real 2-by-2 matrices A0 = � a b c d � , A1 = � a 1 r1b r1c d1 � ,..., AK = � a K rKb rKc dK � , and byρ(M) it stands for the spectral radius of a square matrix M. In this paper, we first show that if bc > 0 then A may be simultaneously symmetrized. This then implies that if bc≥ 0, max{ρ(A0),ρ(A1),...,ρ(AK)} = sup n≥1 max∈ A n n p ρ(M);