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

For a real (n×n)-matrix A the sign-real and the sign-complex spectral radius - invented by Rump - are respectively defined asρR(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)≥ζnρC(A) where the constant ζn:=1−cos⁡(π/n)1+cos⁡(π/n) is best possible.