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.

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 Geometric mean of partial positive definite matrices with missing entries

2019 ◽  
Vol 68 (12) ◽  
pp. 2408-2433
Author(s):  
Hayoung Choi ◽  
Sejong Kim ◽  
Yuanming Shi
Keyword(s):  
Geometric Mean ◽  
Positive Definite ◽  
Positive Definite Matrices

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}.

scholarly journals An Algorithm for Computing Geometric Mean of Two Hermitian Positive Definite Matrices via Matrix Sign

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
F. Soleymani ◽  
M. Sharifi ◽  
S. Shateyi ◽  
F. Khaksar Haghani
Keyword(s):  
Fast Algorithm ◽  
Computational Algorithm ◽  
Geometric Mean ◽  
Positive Definite ◽  
Convergence Order ◽  
Square Root ◽  
Positive Definite Matrices ◽  
Matrix Square Root

Using the relation between a principal matrix square root and its inverse with the geometric mean, we present a fast algorithm for computing the geometric mean of two Hermitian positive definite matrices. The algorithm is stable and possesses a high convergence order. Some experiments are included to support the proposed computational algorithm.

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