Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Some Bounds on Eigenvalues of the Hadamard Product and the Fan Product of Matrices

Version 1 : Received: 14 November 2018 / Approved: 15 November 2018 / Online: 15 November 2018 (15:34:44 CET)

A peer-reviewed article of this Preprint also exists.

Guo, Q.; Leng, J.; Li, H.; Cattani, C. Some Bounds on Eigenvalues of the Hadamard Product and the Fan Product of Matrices. Mathematics 2019, 7, 147. Guo, Q.; Leng, J.; Li, H.; Cattani, C. Some Bounds on Eigenvalues of the Hadamard Product and the Fan Product of Matrices. Mathematics 2019, 7, 147.

Journal reference: Mathematics 2019, 7, 147
DOI: 10.3390/math7020147

## Abstract

In this paper, some mixed type bounds on the spectral radius $\rho(A\circ B)$ for the Hadamard product of two nonnegative matrices ($A$ and $B$) and the minimum eigenvalue $\tau(C\star D)$ of the Fan product of two $M$-matrices ($C$ and $D$) are researched. These bounds complement some corresponding results on the simple type bounds. In addition, a new lower bound on the minimum eigenvalue of the Fan product of several $M$-matrices is also presented: $$\tau(A_{1}\star A_{2}\cdots\star A_{m})\geq \min_{1\leq i\leq n}\{\prod^{m}_{k=1}A_{k}(i,i)-\prod^{m}_{k=1}[A_{k}(i,i)^{P_{k}}-\tau(A_{k}^{(P_{k})})]^\frac{1}{P_{k}}\},$$ where $A_{1},\ldots, A_{k}$ are $n\times n$ $M$-matrices and $P_{1},\ldots, P_{k}>0$ satisfy $\sum^{m}_{k=1}\frac{1}{P_{k}}\geq 1$. Some special cases of the above result and numerical examples show that this new bound improves some existing results.

## Subject Areas

Hadamard product; Nonnegative matrices; Spectral radius; Fan product; M-matrix; Inverse $M$-matrix; Minimum eigenvalue

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