Banach Journal of Mathematical Analysis

On the spectral radius of Hadamard products of nonnegative matrices

Dongjun Chen and Yun Zhang

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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 $$ \rho(A_{1}\circ A_{2} \circ \cdots \circ A_{k}) \leq [\rho(A_{1}A_2\cdots A_{k})]^{1-\frac{2}{k}}[\rho((A_{1}\circ A_{1})\cdots (A_{k}\circ A_{k})]^{\frac{1}{k}} \leq \rho(A_{1}A_2 \cdots A_{k}).$$

Article information

Banach J. Math. Anal., Volume 9, Number 2 (2015), 127-133.

First available in Project Euclid: 19 December 2014

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Zentralblatt MATH identifier

Primary: 15A18: Eigenvalues, singular values, and eigenvectors
Secondary: 47A10: Spectrum, resolvent 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Hadamard product nonnegative matrices spectral radius


Chen, Dongjun; Zhang, Yun. On the spectral radius of Hadamard products of nonnegative matrices. Banach J. Math. Anal. 9 (2015), no. 2, 127--133. doi:10.15352/bjma/09-2-10.

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