Some inequalities for the spectral radius of the Hadamard product of two nonnegative matrices

In this paper, an upper bound on the spectral radius ρ ( A ∘ B ) for the Hadamard product of two nonnegative matrices (A and B) and the minimum eigenvalue τ ( C ★ 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. These results and numerical examples show that the new bounds improve some existing results.

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NEW UPPER BOUNDS ON THE SPECTRAL RADIUS OF THE HADAMARD PRODUCT OF NONNEGATIVE MATRICES

JP Journal of Algebra Number Theory and Applications ◽

10.17654/nt048020121 ◽

2020 ◽

Vol 48 (2) ◽

pp. 121-131

Author(s):

Leena Sharma ◽

V. H. Badshah ◽

H. K. Patel

Keyword(s):

Spectral Radius ◽

Hadamard Product ◽

Upper Bounds ◽

Nonnegative Matrices

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

10.20944/preprints201811.0367.v1 ◽

2018 ◽

Author(s):

Qianping Guo ◽

Jinsong Leng ◽

Houbiao Li ◽

Carlo Cattani

Keyword(s):

Lower Bound ◽

Spectral Radius ◽

Mixed Type ◽

Hadamard Product ◽

Simple Type ◽

Nonnegative Matrices ◽

Minimum Eigenvalue ◽

Numerical Examples ◽

Special Cases ◽

Product Of Matrices

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.

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