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 Inequalities for sector matrices and positive linear maps

2019 ◽  
Vol 35 ◽  
pp. 418-423 ◽  
Author(s):  
Fuping Tan ◽  
Huimin Che
Keyword(s):  
Geometric Mean ◽  
Positive Definite ◽  
Norm Inequalities ◽  
Linear Map ◽  
Linear Maps ◽  
Positive Linear Map ◽  
Positive Linear Maps

Ando proved that if A, B are positive definite, then for any positive linear map Φ, it holds Φ(A#λB) ≤ Φ(A)#λΦ(B), where A#λB, 0 ≤ λ ≤ 1, means the weighted geometric mean of A, B. Using the recently defined geometric mean for accretive matrices, Ando’s result is extended to sector matrices. Some norm inequalities are considered as well.    

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.

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,.

scholarly journals Improving some operator inequalities for positive linear maps

Filomat ◽  
2018 ◽  
Vol 32 (12) ◽  
pp. 4333-4340 ◽  
Author(s):  
Chaojun Yang ◽  
Fangyan Lu
Keyword(s):  
Operator Inequalities ◽  
Linear Map ◽  
Linear Maps ◽  
Operator Version ◽  
Positive Linear Maps

Let 0 < mI ? A ? m'I ? M'I ? B ? MI and p ? 1. Then for every positive unital linear map ?, ?2p(A?tB)?(K(h,2)/41p-1(1+Q(t)(log M'm')2) 2p?2p(A#tB) and ?2p(A?tB)?(K(h,2)/41p-1(1+Q(t)(logM'm')2) 2p(?(A)#t ?(B))2p, where t ? [0,1], h = M/m, K(h,2) = (h+1)2/4h, Q(t) = t2/2(1-t/t)2t and Q(0) = Q(1) = 0. Moreover, we give an improvement for the operator version ofWielandt inequality.

scholarly journals Positive linear maps on normal matrices

2018 ◽  
Vol 29 (12) ◽  
pp. 1850088 ◽  
Author(s):  
Jean-Christophe Bourin ◽  
Eun-Young Lee
Keyword(s):  
Normal Matrix ◽  
Linear Combinations ◽  
Linear Map ◽  
Normal Matrices ◽  
Schur Product ◽  
Linear Maps ◽  
Positive Linear Map ◽  
Matrix Formula ◽  
Positive Linear Maps ◽  
Unitary Orbit

For a positive linear map [Formula: see text] and a normal matrix [Formula: see text], we show that [Formula: see text] is bounded by some simple linear combinations in the unitary orbit of [Formula: see text]. Several elegant sharp inequalities are derived, for instance for the Schur product of two normal matrices [Formula: see text], [Formula: see text] for some unitary [Formula: see text], where the constant [Formula: see text] is optimal.

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.

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).

Facial Structures for the Positive Linear Maps Between Matrix Algebras

1996 ◽  
Vol 39 (1) ◽  
pp. 74-82 ◽  
Author(s):  
Seung-Hyeok Kye
Keyword(s):  
Complete Lattice ◽  
Matrix Algebra ◽  
Convex Set ◽  
Matrix Algebras ◽  
Linear Maps ◽  
The Matrix ◽  
Positive Linear Maps

AbstractLet denote the convex set of all positive linear maps from the matrix algebra Mn(ℂ) into itself. We construct a join homomorphism from the complete lattice of all faces of into the complete lattice of all join homomorphisms between the lattice of all subspaces of ℂn . We also characterize all maximal faces of .

scholarly journals Some operator inequalities for operator means and positive linear maps

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3751-3758
Author(s):  
Jianguo Zhao
Keyword(s):  
Type Inequality ◽  
Real Numbers ◽  
Positive Real ◽  
Operator Inequalities ◽  
Linear Map ◽  
Operator Mean ◽  
Linear Maps ◽  
Operator Means ◽  
Arbitrary Operator ◽  
Positive Linear Maps

In this note, some operator inequalities for operator means and positive linear maps are investigated. The conclusion based on operator means is presented as follows: Let ? : B(H) ? B(K) be a strictly positive unital linear map and h-1 IH ? A ? h1IH and h-12 IH ? B ? h2IH for positive real numbers h1, h2 ? 1. Then for p > 0 and an arbitrary operator mean ?, (?(A)??(B))p ? ?p?p(A?*B), where ?p = max {?2(h1,h2)/4)p, 1/16?2p(h1,h2)}, ?(h1h2) = (h1 + h-1 1)?(h2 + h-12). Likewise, a p-th (p ? 2) power of the Diaz-Metcalf type inequality is also established.

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