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.