Linear operators preserving the sign-real spectral radius Original Research Article

Define the sign-real spectral radius of a real n×n matrix A as image , where ρ0(A)=max{midλmid;λ a real eigenvalue of A} is the real spectral radius of A and image denotes the set of signature matrices, i.e. image , the absolute value of matrices being meant entrywise. In this paper we show that linear invertible operators F on the space of n×n real matrices image preserving the sign-real spectral radius are exactly the operators of the form F(A)=PTD−1SA(T)DP with P a permutation matrix, D a diagonal matrix and S a signature matrix.