scholarly journals A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices

2018 ◽  
Vol 16 (1) ◽  
pp. 298-310
Author(s):  
Xiaoxiao Wang ◽  
Chaoqian Li ◽  
Yaotang Li
Keyword(s):  
Linear Algebra ◽  
Complex Plane ◽  
Upper Bound ◽  
Stochastic Matrix ◽  
Multilinear Algebra ◽  
Stochastic Matrices ◽  
Eigenvalue Localization ◽  
Localization Set

AbstractA set in the complex plane which involves n parameters in [0, 1] is given to localize all eigenvalues different from 1 for stochastic matrices. As an application of this set, an upper bound for the moduli of the subdominant eigenvalues of a stochastic matrix is obtained. Lastly, we fix n parameters in [0, 1] to give a new set including all eigenvalues different from 1, which is tighter than those provided by Shen et al. (Linear Algebra Appl. 447 (2014) 74-87) and Li et al. (Linear and Multilinear Algebra 63(11) (2015) 2159-2170) for estimating the moduli of subdominant eigenvalues.

An Elementary Proof of Johnson–Dulmage–Mendelsohn's Refinement of Birkhoff's Theorem on Doubly Stochastic Matrices

1979 ◽  
Vol 22 (1) ◽  
pp. 81-86 ◽  
Author(s):  
Akihiro Nishi
Keyword(s):  
Upper Bound ◽  
Elementary Proof ◽  
Convex Combination ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices ◽  
Birkhoff’S Theorem

SummaryA purely combinatorial and elementary proof of Johnson-Dulmage-Mendelsohn's theorem, which gives a quite sharp upper bound on the number of permutation matrices needed for representing a doubly stochastic matrix by their convex combination, is given.

scholarly journals New Sufficient Condition for the Positive Definiteness of Fourth Order Tensors

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 303 ◽  
Author(s):  
Jun He ◽  
Yanmin Liu ◽  
Junkang Tian ◽  
Zhuanzhou Zhang
Keyword(s):  
Spectral Radius ◽  
Fourth Order ◽  
Upper Bound ◽  
Positive Definiteness ◽  
Sufficient Condition ◽  
Numerical Examples ◽  
Eigenvalue Localization ◽  
Weakly Symmetric ◽  
Localization Set

In this paper, we give a new Z-eigenvalue localization set for Z-eigenvalues of structured fourth order tensors. As applications, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative fourth order tensors is obtained and a new Z-eigenvalue based sufficient condition for the positive definiteness of fourth order tensors is also presented. Finally, numerical examples are given to verify the efficiency of our results.

scholarly journals The Inequalities of Merris and Foregger for Permanents

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1782
Author(s):  
Divya K. Udayan ◽  
Kanagasabapathi Somasundaram
Keyword(s):  
Symmetric Group ◽  
Linear Algebra ◽  
Symmetric Function ◽  
Stochastic Matrix ◽  
Elementary Symmetric Function ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Classes Of Matrices ◽  
The Matrix ◽  
Element Set

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.

scholarly journals On an Algorithm of G. Birkhoff Concerning Doubly Stochastic Matrices

1960 ◽  
Vol 3 (3) ◽  
pp. 237-242 ◽  
Author(s):  
Diane M. Johnson ◽  
A. L. Dulmage ◽  
N. S. Mendelsohn
Keyword(s):  
Convex Hull ◽  
Upper Bound ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Von Neumann ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices ◽  
Set Of Points

In [1] G. Birkhoff stated an algorithm for expressing a doubly stochastic matrix as an average of permutation matrices. In this note we prove two graphical lemmas and use these to find an upper bound for the number of permutation matrices which the Birkhoff algorithm may use.A doubly stochastic matrix is a matrix of non-negative elements with row and column sums equal to unity and is there - fore a square matrix. A permutation matrix is an n × n doubly stochastic matrix which has n2-n zeros and consequently has n ones, one in each row and one in each column. It has been shown by Birkhoff [1],Hoffman and Wielandt [5] and von Neumann [7] that the set of all doubly stochastic matrices, considered as a set of points in a space of n2 dimensions constitute the convex hull of permutation matrices.

scholarly journals Two new eigenvalue localization sets for tensors and theirs applications

2017 ◽  
Vol 15 (1) ◽  
pp. 1267-1276 ◽  
Author(s):  
Jianxing Zhao ◽  
Caili Sang
Keyword(s):  
Linear Algebra ◽  
Symmetric Tensor ◽  
Sufficient Condition ◽  
Numerical Examples ◽  
Eigenvalue Localization ◽  
Even Order ◽  
Weakly Symmetric ◽  
Localization Set ◽  
Theoretical Results ◽  
Localization Sets

Abstract A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324) and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50). As an application, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1), 187-198). As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

scholarly journals Improved Brauer-type eigenvalue localization sets for tensors with their applications

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4607-4625
Author(s):  
Zhengge Huang ◽  
Jingjing Cui
Keyword(s):  
Linear Algebra ◽  
Multilinear Algebra ◽  
Numerical Examples ◽  
Eigenvalue Localization ◽  
New Criteria ◽  
Localization Sets ◽  
Better Than

In this paper, by excluding some sets from the Brauer-type eigenvalue inclusion sets for tensors developed by Bu et al. (Linear Algebra Appl. 512 (2017) 234-248) and Li et al. (Linear and Multilinear Algebra 64 (2016) 727-736), some improved Brauer-type eigenvalue localization sets for tensors are given, which are proved to be much tighter than those put forward by Bu et al. and Li et al. As applications, some new criteria for identifying the nonsingularity of tensors are developed, which are better than some previous results. This fact is illustrated by some numerical examples.

scholarly journals A modified S-type eigenvalue localization set of tensors applications

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6395-6416
Author(s):  
Zhengge Huang ◽  
Ligong Wang ◽  
Zhong Xu ◽  
Jingjing Cui
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Numerical Examples ◽  
Nonnegative Tensors ◽  
Eigenvalue Localization ◽  
Localization Set ◽  
Weakly Irreducible ◽  
Eigenvalues Of Tensors

Based on the S-type eigenvalue localization set developed by Li et al. (Linear Algebra Appl. 493 (2016) 469-483) for tensors, a modified S-type eigenvalue localization set for tensors is established in this paper by excluding some sets from the existing S-type eigenvalue localization set developed by Huang et al. (arXiv: 1602.07568v1, 2016). The proposed set containing all eigenvalues of tensors is much sharper compared with that employed by Li et al. and Huang et al. As its applications, a criteria, which can be utilized for identifying the nonsingularity of tensors, is developed. In addition, we provide new upper and lower bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of weakly irreducible strong M-tensors. These bounds are superior to some previous results, which is illustrated by some numerical examples.

Permanents of Random Doubly Stochastic Matrices

1974 ◽  
Vol 26 (3) ◽  
pp. 600-607 ◽  
Author(s):  
R. C. Griffiths
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Survey Article ◽  
Doubly Stochastic Matrices ◽  
Image Position

The permanent of an n × n matrix A = (aij) is defined aswhere Sn is the symmetric group of order n. For a survey article on permanents the reader is referred to [2]. An unresolved conjecture due to van der Waerden states that if A is an n × n doubly stochastic matrix; then per (A) ≧ n!/nn, with equality if and only if A = Jn = (1/n).

Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

1996 ◽  
Vol 33 (04) ◽  
pp. 974-985 ◽  
Author(s):  
F. Simonot ◽  
Y. Q. Song
Keyword(s):  
Convergence Rate ◽  
Generating Function ◽  
Convergence Rates ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Input Sequence ◽  
Transition Probability Matrix ◽  
Stochastic Matrices ◽  
Preceding Result

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn , where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and π n for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of π n to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r 0 > 1 with , then the exact convergence rate of π n to π is characterized by r 0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between π n and π based on the moments of A.

scholarly journals An S-type eigenvalue localization set for tensors

2016 ◽  
Vol 493 ◽  
pp. 469-483 ◽  
Author(s):  
Chaoqian Li ◽  
Aiquan Jiao ◽  
Yaotang Li
Keyword(s):  
Eigenvalue Localization ◽  
Localization Set
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