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

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 $sumnolimits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } $ we derive a lower bound of the form $$rho left( {sumlimits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } } right) geqslant sumlimits_{l = 1}^k {beta _{lrho } (A^{(l)} ),{text{ }}beta _l >0.} $$ Also we prove that, for arbitrary irreducible nonnegative matrices $A^{{text{ (}}l{text{)}}} (l = 1, ldots ,k),$ , $$rho left( {sumlimits_{l = 1}^k {A^{(l)} } } right) geqslant sumlimits_{l = 1}^k {alpha _{lrho } (A^{(l)} ),} $$ 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.