The sign-real spectral radius and cycle products Original Research Article

The extension of the Perron-Frobenius theory to real matrices without sign restriction uses the sign-real spectral radius as the generalization of the Perron root. The theory was used to extend and solve the conjecture in the affirmative that an ill-conditioned matrix is nearby a singular matrix also in the componentwise sense. The proof estimates the ratio between the sign-real spectral radius and the maximum geometric mean of a cycle product. In this note we discuss bounds for this ratio including a counterexample to a conjecture about this ratio.