Nonnegative matrices, zero patterns, and spectral inequalities (Q1262926)

The theme is inequalities for the spectral radius (\(\rho)\) of a positive linear combination of nonnegative matrices. Here are two of the results. Theorem 3.1: \(\rho (A+sgn(A))\geq \rho (A)+sgn(\rho (A)),\) where sgn is the signum function, applied to a matrix elementwise. Theorem 2.1: Let \(A_ i\) be irreducible and \(t_ i\geq 0\) for \(i=1,...,p\). Let \(P_ i\) be the projector of \(A_ i\) corresponding to its Perron-Frobenius eigenvalue \(\rho (A_ i)\). If \(P_ i-P_ j\) has 0 diagonal for all i and j, then \(\rho (\Sigma t_ iA_ i)\geq \Sigma t_ i\rho (A_ i).\) This theorem has a partial corollary.