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 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.

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.

Bounds for Characteristic Values of Positive Definite Matrices

1972 ◽  
Vol 15 (1) ◽  
pp. 51-56 ◽  
Author(s):  
P. A. Binding ◽  
W. D. Hoskins ◽  
P. J. Ponzo
Keyword(s):  
Related Problem ◽  
Geometric Mean ◽  
General Setting ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Arithmetic Mean ◽  
Positive Real ◽  
Positive Definite Matrices ◽  
Smallest Eigenvalue ◽  
Characteristic Values

We consider the problem of determining the best possible bounds on the eigenvalues of an nth order positive definite matrix B, when the determinant (D) and trace (T) are given. A large variety of bounds on the eigenvalues are known when different information concerning B is available (see, for example, [1], [2]). Since D and T simply provide the geometric mean and arithmetic mean of the positive, real eigenvalues of B, the solution to the problem involves certain inequalities satisfied by these means (see [3] for such inequalities in a more general setting). A related problem in which the largest and smallest eigenvalue are known, and inequalities involving D and T are obtained, is described in [4].

Sign in / Sign up

Export Citation Format

Share Document