On the max version of the generalized spectral radius theorem

Let ΨΨ be a bounded set of n×nn×n non-negative matrices. Recently, the max algebra version μ(Ψ)μ(Ψ) of the generalized spectral radius of ΨΨ was introduced. We show thatμ(Ψ)=limt→∞i(t))1/t,where ρρ denotes the generalized spectral radius and Ψ(t)Ψ(t) the Hadamard power of ΨΨ. This provides a description of μ(Ψ)μ(Ψ) that uses no max terminology. As an application we give a short proof of the max version of the generalized spectral radius theorem.