Upper bound for the non-maximal eigenvalues of irreducible nonnegative matrices

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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An Efficient Algorithm for Finding the Maximal Eigenvalue of Zero Symmetric Nonnegative Matrices

Mathematical Problems in Engineering ◽

10.1155/2018/6438106 ◽

2018 ◽

Vol 2018 ◽

pp. 1-7

Author(s):

Gang Wang ◽

Lihong Sun

Keyword(s):

Efficient Algorithm ◽

Positive Definiteness ◽

Numerical Results ◽

Nonnegative Matrices ◽

Maximal Eigenvalue ◽

Positive Semidefiniteness ◽

Maximal Eigenvalues ◽

Modified Algorithm

In this paper, we propose an improved power algorithm for finding maximal eigenvalues. Without any partition, we can get the maximal eigenvalue and show that the modified power algorithm is convergent for zero symmetric reducible nonnegative matrices. Numerical results are reported to demonstrate the effectiveness of the modified power algorithm. Finally, a modified algorithm is proposed to test the positive definiteness (positive semidefiniteness) of Z-matrices.

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Sharp Bounds on the Spectral Radius of Nonnegative Matrices and Comparison to the Frobenius’ Bounds

International Journal of Circuits, Systems and Signal Processing ◽

10.46300/9106.2020.14.57 ◽

2020 ◽

Vol 14 ◽

Keyword(s):

Lower Bound ◽

Spectral Radius ◽

Discrete Time ◽

Upper Bound ◽

Nonnegative Matrix ◽

Nonnegative Matrices ◽

Sharp Bounds ◽

Similarity Transformations ◽

Linear Invariant ◽

The Stability

In this paper, a new upper bound and a new lower bound for the spectral radius of a nοnnegative matrix are proved by using similarity transformations. These bounds depend only on the elements of the nonnegative matrix and its row sums and are compared to the well-established upper and lower Frobenius’ bounds. The proposed bounds are always sharper or equal to the Frobenius’ bounds. The conditions under which the new bounds are sharper than the Frobenius' ones are determined. Illustrative examples are also provided in order to highlight the sharpness of the proposed bounds in comparison with the Frobenius’ bounds. An application to linear invariant discrete-time nonnegative systems is given and the stability of the systems is investigated. The proposed bounds are computed with complexity O(n2).

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The Synchronizing Probability Function for Primitive Sets of Matrices

International Journal of Foundations of Computer Science ◽

10.1142/s0129054120410051 ◽

2020 ◽

Vol 31 (06) ◽

pp. 777-803

Author(s):

Costanza Catalano ◽

Raphaël M. Jungers

Keyword(s):

Convex Optimization ◽

Upper Bound ◽

Probability Function ◽

Numerical Results ◽

Optimization Techniques ◽

Nonnegative Matrices ◽

Synchronization Process ◽

New Class ◽

Primitive Sets

Motivated by recent results relating synchronizing DFAs and primitive sets, we tackle the synchronization process and the related longstanding Černý conjecture by studying the primitivity phenomenon for sets of nonnegative matrices having neither zero-rows nor zero-columns. We formulate the primitivity process in the setting of a two-player probabilistic game and we make use of convex optimization techniques to describe its behavior. We develop a tool for approximating and upper bounding the exponent of any primitive set and supported by numerical results we state a conjecture that, if true, would imply a quadratic upper bound on the reset threshold of a new class of automata.

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

Mathematics ◽

10.3390/math7020147 ◽

2019 ◽

Vol 7 (2) ◽

pp. 147

Author(s):

Qianping Guo ◽

Jinsong Leng ◽

Houbiao Li ◽

Carlo Cattani

Keyword(s):

Lower Bound ◽

Spectral Radius ◽

Upper Bound ◽

Hadamard Product ◽

Simple Type ◽

Nonnegative Matrices ◽

Minimum Eigenvalue ◽

Numerical Examples ◽

Product Of 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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The Algebraic Degree of Phase-Type Distributions

Journal of Applied Probability ◽

10.1239/jap/1285335399 ◽

2010 ◽

Vol 47 (3) ◽

pp. 611-629 ◽

Cited By ~ 2

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.

Download Full-text