A short note on the ratio between sign-real and sign-complex spectral radius of a real square matrix

Abstract

For a real (n×n)-matrix A the sign-real and the sign-complex spectral radius – invented by Rump – are respectively defined as ρR(A):=max⁡|λ|||Ax|=|λx|,λ∈R,x∈Rn{0,ρC(A):=max⁡|λ|||Ax|=|λx|,λ∈C,x∈Cn{0. For n≥2 we prove ρR(A)≥ζn ρC(A) where the constant ζn:=[formula omitted] is best possible.