Theorems of Perron-Frobenius type for matrices without sign restrictions (Q1372955)

For \(A\in M_n(R)\) the author defines the `sign-real spectral radius' \(\rho_0^S (A)= \max_{S\in {\mathcal S}} \rho_0 (SA)\), where \(S\) is an \(n\times n\) signature matrix (a diagonal matrix with diagonal entries 1 or \(-1)\) and \(\rho_0(A)\) is the real spectral radius (an absolutely largest real eigenvalue of \(A)\). He derives various properties, characterizations, and bounds of \(\rho_0^S (A)\); for instance, a behavior similar to that of the Perron eigenvalue of a nonnegative matrix. \(\rho_0^S(A)\) is closely related to the componentwise distance to the nearest singular matrix as well as to the Perron eigenvalue of the (entrywise) absolute value of \(A\) and to the \(\mu\)-number.