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 On a Factorisation of Positive Definite Matrices

1963 ◽  
Vol 6 (3) ◽  
pp. 405-407 ◽  
Author(s):  
Kulendra N. Majindar
Keyword(s):  
Positive Definite Matrix ◽  
Positive Definite ◽  
Symmetric Positive Definite Matrix ◽  
Positive Definite Matrices ◽  
Schur Product ◽  
Symmetric Positive Definite ◽  
Symmetric Positive Definite Matrices

All our matrices are square with real elements. The Schur product of two n × n matrices B = (bij) and C = (cij) (i, j, = 1, 2, …, n), is an n × n matrix A = (aij) with aij = bij cij, (i, j = 1, 2, …, n).A result due to Schur [1] states that if B and C are symmetric positive definite matrices then so is their Schur product A. A question now a rises. Can any symmetric positive definite matrix be expressed as a Schur product of two symmetric positive definite matrices? The answer is in the affirmative as we show in the following theorem.

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.

scholarly journals EEG representation using multi-instance framework on the manifold of symmetric positive definite matrices

2019 ◽  
Vol 16 (3) ◽  
pp. 036016 ◽  
Author(s):  
Khadijeh Sadatnejad ◽  
Mohammad Rahmati ◽  
Reza Rostami ◽  
Reza Kazemi ◽  
Saeed S Ghidary ◽  
...  
Keyword(s):  
Positive Definite ◽  
Positive Definite Matrices ◽  
Symmetric Positive Definite ◽  
Symmetric Positive Definite Matrices

scholarly journals A GPU solver for symmetric positive-definite matrices vs. traditional codes

2019 ◽  
Vol 78 (9) ◽  
pp. 2933-2943 ◽  
Author(s):  
Jan Bohacek ◽  
Abdellah Kharicha ◽  
Andreas Ludwig ◽  
Menghuai Wu ◽  
Tobias Holzmann ◽  
...  
Keyword(s):  
Positive Definite ◽  
Positive Definite Matrices ◽  
Symmetric Positive Definite ◽  
Symmetric Positive Definite Matrices

scholarly journals Geometry-aware manipulability learning, tracking, and transfer

2020 ◽  
pp. 027836492094681
Author(s):  
Noémie Jaquier ◽  
Leonel Rozo ◽  
Darwin G Caldwell ◽  
Sylvain Calinon
Keyword(s):  
Gaussian Mixture ◽  
Positive Definite ◽  
Robotic Hand ◽  
Redundant Manipulators ◽  
Positive Definite Matrices ◽  
Symmetric Positive Definite ◽  
Task Requirements ◽  
Robot Performance ◽  
Symmetric Positive Definite Matrices ◽  
Dual Arm

Body posture influences human and robot performance in manipulation tasks, as appropriate poses facilitate motion or the exertion of force along different axes. In robotics, manipulability ellipsoids arise as a powerful descriptor to analyze, control, and design the robot dexterity as a function of the articulatory joint configuration. This descriptor can be designed according to different task requirements, such as tracking a desired position or applying a specific force. In this context, this article presents a novel manipulability transfer framework, a method that allows robots to learn and reproduce manipulability ellipsoids from expert demonstrations. The proposed learning scheme is built on a tensor-based formulation of a Gaussian mixture model that takes into account that manipulability ellipsoids lie on the manifold of symmetric positive-definite matrices. Learning is coupled with a geometry-aware tracking controller allowing robots to follow a desired profile of manipulability ellipsoids. Extensive evaluations in simulation with redundant manipulators, a robotic hand and humanoids agents, as well as an experiment with two real dual-arm systems validate the feasibility of the approach.

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