scholarly journals The densities and distributions of the largest eigenvalue and the trace of a Beta–Wishart matrix

2019 ◽  
Vol 10 (01) ◽  
pp. 2150010
Author(s):  
Vesselin Drensky ◽  
Alan Edelman ◽  
Tierney Genoar ◽  
Raymond Kan ◽  
Plamen Koev
Keyword(s):  
Series Expansions ◽  
Wishart Matrix ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
The Largest Eigenvalue

We present new expressions for the densities and distributions of the largest eigenvalue and the trace of a Beta–Wishart matrix. The series expansions for these expressions involve fewer terms than previously known results. For the trace, we also present a new algorithm that is linear in the size of the matrix and the degree of truncation, which is optimal.

scholarly journals Łojasiewicz-type inequalities with explicit exponents for the largest eigenvalue function of real symmetric polynomial matrices

2016 ◽  
Vol 27 (02) ◽  
pp. 1650012 ◽  
Author(s):  
Si Tiep Dinh ◽  
Tien Son Pham
Keyword(s):  
Distance Function ◽  
Polynomial Matrix ◽  
Symmetric Polynomial ◽  
Clarke Subdifferential ◽  
Polynomial Matrices ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Matrix Formula ◽  
The Clarke Subdifferential ◽  
The Largest Eigenvalue

Let [Formula: see text] be a real symmetric polynomial matrix of order [Formula: see text] and let [Formula: see text] be the largest eigenvalue function of the matrix [Formula: see text] We denote by [Formula: see text] the Clarke subdifferential of [Formula: see text] at [Formula: see text] In this paper, we first give the following nonsmooth version of Łojasiewicz gradient inequality for the function [Formula: see text] with an explicit exponent: For any [Formula: see text] there exist [Formula: see text] and [Formula: see text] such that we have for all [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] is a function introduced by D’Acunto and Kurdyka: [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text] Then we establish some local and global versions of Łojasiewicz inequalities which bound the distance function to the set [Formula: see text] by some exponents of the function [Formula: see text].

scholarly journals On the Spectrum of Laplacian Matrix

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Akbar Jahanbani ◽  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
The Matrix ◽  
Vertex Degrees ◽  
The Largest Eigenvalue

Let G be a simple graph of order n . The matrix ℒ G = D G − A G is called the Laplacian matrix of G , where D G and A G denote the diagonal matrix of vertex degrees and the adjacency matrix of G , respectively. Let l 1 G , l n − 1 G be the largest eigenvalue, the second smallest eigenvalue of ℒ G respectively, and λ 1 G be the largest eigenvalue of A G . In this paper, we will present sharp upper and lower bounds for l 1 G and l n − 1 G . Moreover, we investigate the relation between l 1 G and λ 1 G .

scholarly journals An orthogonal power method of solving the partial eigenproblem for a symmetric nonnegative definite matrix

2016 ◽  
pp. 44-54
Author(s):  
И.В. Киреев
Keyword(s):  
Homogeneous System ◽  
Power Method ◽  
Algebraic Equations ◽  
Largest Eigenvalue ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Conjugate Direction Method ◽  
Nonnegative Definite ◽  
The Largest Eigenvalue ◽  
Square Matrix

Предложена и обоснована экономичная версия метода сопряженных направлений для построения нетривиального решения однородной системы линейных алгебраических уравнений с вырожденной симметричной неотрицательно определенной квадратной матрицей. Предложено однопараметрическое семейство одношаговых нелинейных итерационных процессов вычисления собственного вектора, отвечающего наибольшему собственному значению симметричной неотрицательно определенной квадратной матрицы. Это семейство включает в себя степенной метод как частный случай. Доказана сходимость возникающих последовательностей векторов к собственному вектору, ассоциированному с наибольшим характеристическим числом матрицы. Предложена двухшаговая процедура ускорения сходимости итераций этих процессов, в основе которой лежит ортогонализация в подпространстве Крылова. Приведены результаты численных экспериментов. An efficient version of the conjugate direction method to find a nontrivial solution of a homogeneous system of linear algebraic equations with a singular symmetric nonnegative definite square matrix is proposed and substantiated. A one-parameter family of one-step nonlinear iterative processes to determine the eigenvector corresponding to the largest eigenvalue of a symmetric nonnegative definite square matrix is also proposed. This family includes the power method as a special case. The convergence of corresponding vector sequences to the eigenvector associated with the largest eigenvalue of the matrix is proved. A two-step procedure is formulated to accelerate the convergence of iterations for these processes. This procedure is based on the orthogonalization in Krylov subspaces. A number of numerial results are discussed.

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