The general Hermitian nonnegative-definite and positive-definite solutions to the matrix equationGXG *+HYH *=C

2004 ◽  
Vol 14 (1-2) ◽  
pp. 51-67 ◽  
Author(s):  
Xian Zhang
Keyword(s):  
Positive Definite ◽  
The Matrix ◽  
Nonnegative Definite

On Projection of a Positive Definite Matrix on a Cone of Nonnegative Definite Toeplitz Matrices

2018 ◽  
Vol 33 ◽  
pp. 74-82 ◽  
Author(s):  
Katarzyna Filipiak ◽  
Augustyn Markiewicz ◽  
Adam Mieldzioc ◽  
Aneta Sawikowska
Keyword(s):  
Toeplitz Matrices ◽  
Positive Definite Matrix ◽  
Positive Definiteness ◽  
Positive Definite ◽  
Asymptotic Cone ◽  
The Matrix ◽  
Multi Level ◽  
Nonnegative Definite ◽  
Unknown Covariance ◽  
Numer Math

We consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be applied in statistics, for example in the estimation of unknown covariance structures under the multi-level multivariate models, where positive definiteness is required. We conduct simulation studies to compare statistical properties of the estimators obtained by projection on the cone with a given matrix dimension and on the asymptotic cone.

scholarly journals The common Re-nonnegative definite and Re-positive definite solutions to the matrix equations $ A_1XA_1^\ast = C_1 $ and $ A_2XA_2^\ast = C_2 $

2021 ◽  
Vol 7 (1) ◽  
pp. 384-397
Author(s):  
Yinlan Chen ◽  
◽  
Lina Liu
Keyword(s):  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
The Matrix ◽  
The Common ◽  
Necessary And Sufficient ◽  
Nonnegative Definite ◽  
Linear Matrix Equations

<abstract><p>In this paper, we consider the common Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions to a pair of linear matrix equations $ A_1XA_1^\ast = C_1, \ A_2XA_2^\ast = C_2 $ and present some necessary and sufficient conditions for their solvability as well as the explicit expressions for the general common Re-nnd and Re-pd solutions when the consistent conditions are satisfied.</p></abstract>

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.

A Trace Inequality for Symmetric Nonnegative Definite Matrices

2011 ◽  
Vol 225-226 ◽  
pp. 970-973
Author(s):  
Shi Qing Wang
Keyword(s):  
Control Theory ◽  
Communication Systems ◽  
Positive Definite ◽  
Trace Inequality ◽  
Positive Definite Matrices ◽  
Symmetric Positive Definite ◽  
Multiple Input ◽  
Nonnegative Definite ◽  
Symmetric Positive Definite Matrices ◽  
Trace Inequalities

Trace inequalities naturally arise in control theory and in communication systems with multiple input and multiple output. One application of Belmega’s trace inequality has already been identified [3]. In this paper, we extend the symmetric positive definite matrices of his inequality to symmetric nonnegative definite matrices, and the inverse matrices to Penrose-Moore inverse matrices.

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