New lower bound of the determinant for Hadamard product on some totally 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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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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On a Schur complement inequality for the Hadamard product of certain totally nonnegative matrices

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.1436 ◽

2011 ◽

Vol 22 ◽

Cited By ~ 1

Author(s):

Zhongpeng Yang ◽

Xiaoxia Feng

Keyword(s):

Hadamard Product ◽

Schur Complement ◽

Nonnegative Matrices ◽

Totally Nonnegative Matrices

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The Algebraic Degree of Phase-Type Distributions

Journal of Applied Probability ◽

10.1017/s0021900200006963 ◽

2010 ◽

Vol 47 (03) ◽

pp. 611-629

Author(s):

Mark Fackrell ◽

Qi-Ming He ◽

Peter Taylor ◽

Hanqin Zhang

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Polynomial Algorithm ◽

Algebraic Degree ◽

Nonnegative Matrices ◽

Stieltjes Transform ◽

Special Cases ◽

Phase Type ◽

Phase Type Distributions ◽

The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

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