Lower Bounds for the Perron Root of a Sum of Nonnegative Matrices

Abstract

Let \(A^{(l)} (l = 1, \ldots ,k)\) be \(n \times n\) nonnegative matrices with right and left Perron vectors \(u^{(l)} \) and \(v^{(l)} \), respectively, and let \(D^{(l)} \) and \(E^{(l)} (l = 1, \ldots ,k)\) be positive-definite diagonal matrices of the same order. Extending known results, under the assumption that $$u^{(1)} \circ v^{(1)} = \ldots = u^{(k)} \circ v^{(k)} \ne 0$$ (where ``\( \circ \)'' denotes the componentwise, i.e., the Hadamard product of vectors) but without requiring that the matrices \(A^{(l)} \) be irreducible, for the Perron root of the sum \(\sum\nolimits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } \) we derive a lower bound of the form $$\rho \left( {\sum\limits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } } \right) \geqslant \sum\limits_{l = 1}^k {\beta _{l\rho } (A^{(l)} ),{\text{ }}\beta _l >0.} $$ Also we prove that, for arbitrary irreducible nonnegative matrices \(A^{{\text{ (}}l{\text{)}}} (l = 1, \ldots ,k),\), $$\rho \left( {\sum\limits_{l = 1}^k {A^{(l)} } } \right) \geqslant \sum\limits_{l = 1}^k {\alpha _{l\rho } (A^{(l)} ),} $$ where the coefficients ∝1>0 are specified using an arbitrarily chosen normalized positive vector. The cases of equality in both estimates are analyzed, and some other related results are established. Bibliography: 8 titles.