Lower bounds for the Perron root of a~sum of nonnegative matrices

L. Yu. Kolotilina

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIV, Tome 268 (2000), pp. 49-71

Résumé

Let $A^{(l)}$ $(l=1,\dots,k)$ be $n\times n$ nonnegative matrices with right and left Perron vectors $u^{(l)}$ and $v^{(l)}$, respectively, and let $D^{(l)}$ and $E^{(l)}$ $(l=1,\dots,k)$ be positive-definite diagonal matrices of the same order. Extending known results, under the assumption that $$ u^{(1)}\circ v^{(1)}=\dots=u^{(k)}\circ v^{(k)}\ne0 $$ (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 $\sum^k_{l=1}D^{(l)}A^{(l)}E^{(l)}$ we derive a lower bound of the form $$ \rho\left(\sum^k_{l=1}D^{(l)}A^{(l)}E^{(l)}\right)\ge\sum^{k}_{l=1}\beta_l\rho(A^{(l)}),\quad\beta_l>0. $$ Also we prove that, for arbitrary irreducible nonnegative matrices $A^{(l)}$ $(l=1,\ldots,k)$, $$ \rho\left(\sum^{k}_{l=1}A^{(l)}\right)\ge\sum^k_{l=1}\alpha_l\rho(A^{(l)}), $$ where the coefficients $\alpha_l>0$ 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.