Linear operators preserving the sign-real spectral radius (Q1970509)

The sign-real spectral radius of a real \(n \times n\) matrix \(A\) is defined as the quantity \(\rho_0^S(A)=\) max \(\rho_0(SA)\), where \(S\) is a signature matrix i. e. the diagonal matrix with diagonal entries equal to \(+1\) or \(-1\) and \(\rho_0\) is the spectral radius of a real matrix. It is proved that linear invertible operators \(F\) on the real \(n \times n\) matrices which preserve the sign-real radius of the matrix \(A\) have exactly the form \(F(A)=P^TD^{-1}SA^{(T)}DP\), where \(P\) is a permutation matrix, \(D\) is a nonsingular diagonal matrix and \(S\) is a signature matrix.