On the spectral radius of Hadamard products of nonnegative matrices

Abstract

We present some spectral radius inequalities for nonnegative matrices. Using the ideas of Audenaert, we then prove the inequality which may be regarded as a Cauchy--Schwarz inequality for spectral radius of nonnegative matrices $$ \rho(A \circ B) \leq [\rho(A \circ A)]^{\frac{1}{2}}[\rho(B\circ B)]^{\frac{1}{2}}. $$ In addition, new proofs of some related results due to Horn and Zhang, Huang are also given. Finally, we interpolate Huang's inequality by proving $$ \rho(A_{1}\circ A_{2} \circ \cdots \circ A_{k}) \leq [\rho(A_{1}A_2\cdots A_{k})]^{1-\frac{2}{k}}[\rho((A_{1}\circ A_{1})\cdots (A_{k}\circ A_{k})]^{\frac{1}{k}} \leq \rho(A_{1}A_2 \cdots A_{k}).$$