scholarly journals On the Marginal Distribution of the Diagonal Blocks in a Blocked Wishart Random Matrix

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Kjetil B. Halvorsen ◽  
Victor Ayala ◽  
Eduardo Fierro
Keyword(s):  
Random Matrix ◽  
Random Fields ◽  
Spatial Interpolation ◽  
Marginal Distribution ◽  
Positive Definite ◽  
Composite Likelihood ◽  
Marginal Density ◽  
Likelihood Methods ◽  
Positive Definite Matrices ◽  
The Matrix

Let A be a (m1+m2)×(m1+m2) blocked Wishart random matrix with diagonal blocks of orders m1×m1 and m2×m2. The goal of the paper is to find the exact marginal distribution of the two diagonal blocks of A. We find an expression for this marginal density involving the matrix-variate generalized hypergeometric function. We became interested in this problem because of an application in spatial interpolation of random fields of positive definite matrices, where this result will be used for parameter estimation, using composite likelihood methods.

Identifying graph clusters using variational inference and links to covariance parametrization

2009 ◽  
Vol 367 (1906) ◽  
pp. 4407-4426
Author(s):  
David Barber
Keyword(s):  
Mean Field ◽  
Matrix Decomposition ◽  
Difficult Problem ◽  
Positive Definite ◽  
Variational Inference ◽  
Field Theories ◽  
Variational Approximation ◽  
Positive Definite Matrices ◽  
The Matrix ◽  
Non Gaussian

Finding clusters of well-connected nodes in a graph is a problem common to many domains, including social networks, the Internet and bioinformatics. From a computational viewpoint, finding these clusters or graph communities is a difficult problem. We use a clique matrix decomposition based on a statistical description that encourages clusters to be well connected and few in number. The formal intractability of inferring the clusters is addressed using a variational approximation inspired by mean-field theories in statistical mechanics. Clique matrices also play a natural role in parametrizing positive definite matrices under zero constraints on elements of the matrix. We show that clique matrices can parametrize all positive definite matrices restricted according to a decomposable graph and form a structured factor analysis approximation in the non-decomposable case. Extensions to conjugate Bayesian covariance priors and more general non-Gaussian independence models are briefly discussed.

scholarly journals Some matrix power and Karcher means inequalities involving positive linear maps

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2625-2634
Author(s):  
Monire Hajmohamadi ◽  
Rahmatollah Lashkaripour ◽  
Mojtaba Bakherad
Keyword(s):  
Weight Vector ◽  
Positive Definite ◽  
Matrix Inequalities ◽  
Positive Definite Matrices ◽  
Linear Map ◽  
Linear Maps ◽  
Karcher Mean ◽  
The Matrix ◽  
Positive Linear Maps ◽  
Matrix Power

In this paper, we generalize some matrix inequalities involving the matrix power means and Karcher mean of positive definite matrices. Among other inequalities, it is shown that if A = (A1,...,An) is an n-tuple of positive definite matrices such that 0 < m ? Ai ? M (i = 1,...,n) for some scalars m < M and ? = (w1,...,wn) is a weight vector with wi ? 0 and ?n,i=1 wi=1, then ?p (?n,i=1 wiAi)? ?p?p(Pt(?,A)) and ?p (?n,i=1 wiAi) ? ?p?p(?(?,A)), where p > 0,? = max {(M+m)2/4Mm,(M+m)2/42p Mm}, ? is a positive unital linear map and t ? [-1,1]\{0}.

On the Schur product ofH-matrices and non-negative matrices, and related inequalities

1964 ◽  
Vol 60 (3) ◽  
pp. 425-431 ◽  
Author(s):  
M. S. Lynn
Keyword(s):  
Real Line ◽  
Positive Definite ◽  
Positive Definite Matrices ◽  
The Real ◽  
Schur Product ◽  
Symmetric Positive Definite ◽  
The Matrix ◽  
Image Position ◽  
Symmetric Positive Definite Matrices

1.Introduction. Let ℛndenote the set of alln×nmatrices with real elements, and letdenote the subset of ℛnconsisting of all real,n×n, symmetric positive-definite matrices. We shall use the notationto denote that minor of the matrixA= (aij) ∈ ℛnwhich is the determinant of the matrixTheSchur Product(Schur (14)) of two matricesA, B∈ ℛnis denned bywhereA= (aij),B= (bij),C= (cij) andLet ϕ be the mapping of ℛninto the real line defined byfor allA∈ ℛn, where, as in the sequel,.

A note on the matrix arithmetic-geometric mean inequality

2018 ◽  
Vol 34 ◽  
pp. 283-287
Author(s):  
Teng Zhang
Keyword(s):  
Positive Integer ◽  
Geometric Mean ◽  
Positive Definite ◽  
Operator Norm ◽  
Positive Definite Matrices ◽  
The Matrix ◽  
Special Case

This note proves the following inequality: If $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $\bA_1,\bA_2,\dots,\bA_n$, the following inequality holds: \begin{equation*}\label{eq:main} \frac{1}{n^3} \, \Big\|\sum_{j_1,j_2,j_3=1}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\| \,\geq\, \frac{(n-3)!}{n!} \, \Big\|\sum_{\substack{j_1,j_2,j_3=1,\\\text{$j_1$, $j_2$, $j_3$ all distinct}}}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\|, \end{equation*} where $\|\cdot\|$ represents the operator norm. This inequality is a special case of a recent conjecture proposed by Recht and R\'{e} (2012).

scholarly journals Study On Some Matrix Equations Involving The Weighted Geometric Mean and Their Application

10.29007/7sj7 ◽  
2022 ◽  
Author(s):  
Xuan Dai Le ◽  
Tuan Cuong Pham ◽  
Thi Hong Van Nguyen ◽  
Nhat Minh Tran ◽  
Van Vinh Dang
Keyword(s):  
Geometric Mean ◽  
Positive Definite ◽  
Quantum States ◽  
Matrix Equations ◽  
Positive Definite Matrices ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Fidelity Measure ◽  
The Fixed Point Theorem ◽  
Definite Solution

In this paper we consider two matrix equations that involve the weighted geometric mean. We use the fixed point theorem in the cone of positive definite matrices to prove the existence of a unique positive definite solution. In addition, we study the multi-step stationary iterative method for those equations and prove the corresponding convergence. A fidelity measure for quantum states based on the matrix geometric mean is introduced as an application of matrix equation.

M-convolutions of products and ratios, statistical distributions and fractional calculus

Analysis ◽  
2016 ◽  
Vol 36 (1) ◽  
Author(s):  
Arakaparampil M. Mathai
Keyword(s):  
Fractional Calculus ◽  
Random Variables ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Positive Definite Matrices ◽  
The Real ◽  
Real Scalar ◽  
Scalar Variable ◽  
The Matrix

AbstractIt is shown that Mellin convolutions of products and ratios in the real scalar variable case can be considered as densities of products and ratios of two independently distributed real scalar positive random variables. It is also shown that these are also connected to Krätzel integrals and to the Krätzel transform in applied analysis, to reaction-rate probability integrals in astrophysics and to other related aspects when the random variables have gamma or generalized gamma densities, and to fractional calculus when one of the variables has a type-1 beta density and the other variable has an arbitrary density. Matrix-variate analogues are also discussed. In the matrix-variate case, the M-convolutions introduced by the author are shown to be directly connected to densities of products and ratios of statistically independently distributed positive definite matrix random variables in the real case and to Hermitian positive definite matrices in the complex domain. These M-convolutions reduce to Mellin convolutions in the scalar variable case.

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