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 Inequalities for the eigenvalues of the positive definite solutions of the nonlinear matrix equation

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2667-2671
Author(s):  
Guoxing Wu ◽  
Ting Xing ◽  
Duanmei Zhou
Keyword(s):  
Matrix Equation ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
The Matrix ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix ◽  
Positive Integers ◽  
Equivalent Conditions

In this paper, the Hermitian positive definite solutions of the matrix equation Xs + A*X-tA = Q are considered, where Q is a Hermitian positive definite matrix, s and t are positive integers. Bounds for the sum of eigenvalues of the solutions to the equation are given. The equivalent conditions for solutions of the equation are obtained. The eigenvalues of the solutions of the equation with the case AQ = QA are investigated.

scholarly journals X−1-balance of some partially balanced experimental designs with particular emphasis on block and row-column designs

2015 ◽  
Vol 52 (1) ◽  
pp. 1-12
Author(s):  
Ryszard Walkowiak
Keyword(s):  
Information Matrix ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
The Other ◽  
Experimental Designs ◽  
Block Designs ◽  
The Matrix

SummaryThis paper considers block designs and row-column designs where the information matrix C has two different nonzero eigenvalues, one of multiplicity 1 and the other of multiplicity h−1, where h is the rank of the matrix C. It was found that for each such design there exists a diagonal positive definite matrix X such that the design is X −1-balanced.

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>

scholarly journals On the estimation of ${x}^TA^{-1}{x}$ for symmetric matrices

2021 ◽  
Vol 37 ◽  
pp. 549-561
Author(s):  
Paraskevi Fika ◽  
Marilena Mitrouli ◽  
Ondrej Turec
Keyword(s):  
Quadratic Form ◽  
Mathematical Problem ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Symmetric Matrices ◽  
Symmetric Positive Definite Matrix ◽  
Matrix Inverse ◽  
Symmetric Positive Definite ◽  
The Matrix

The central mathematical problem studied in this work is the estimation of the quadratic form $x^TA^{-1}x$ for a given symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ and vector $x \in \mathbb{R}^n$. Several methods to estimate $x^TA^{-1}x$ without computing the matrix inverse are proposed. The precision of the estimates is analyzed both analytically and numerically.  

scholarly journals An Analytic Proof of the Matrix Spectral Factorization Theorem

2008 ◽  
Vol 15 (2) ◽  
pp. 241-249
Author(s):  
Lasha Ephremidze ◽  
Gigla Janashia ◽  
Edem Lagvilava
Keyword(s):  
Positive Definite Matrix ◽  
Positive Definite ◽  
Spectral Factorization ◽  
Factorization Theorem ◽  
Matrix Functions ◽  
Analytic Proof ◽  
Wiener’S Theorem ◽  
The Matrix ◽  
Matrix Spectral Factorization

Abstract An analytic proof of Wiener's theorem on factorization of positive definite matrix-functions is proposed.

Iteration with Stepsize Parameter and Condition Numbers for a Nonlinear Matrix Equation

2018 ◽  
Vol 34 ◽  
pp. 217-230
Author(s):  
Syed M. Raza Shah Naqvi ◽  
Jie Meng ◽  
Hyun-Min Kim
Keyword(s):  
Matrix Equation ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Condition Numbers ◽  
Symmetric Positive Definite Matrix ◽  
Nonlinear Matrix Equation ◽  
Symmetric Positive Definite ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Definite Solution

In this paper, the nonlinear matrix equation $X^p+A^TXA=Q$, where $p$ is a positive integer, $A$ is an arbitrary $n\times n$ matrix, and $Q$ is a symmetric positive definite matrix, is considered. A fixed-point iteration with stepsize parameter for obtaining the symmetric positive definite solution of the matrix equation is proposed. The explicit expressions of the normwise, mixed and componentwise condition numbers are derived. Several numerical examples are presented to show the efficiency of the proposed iterative method with proper stepsize parameter and the sharpness of the three kinds of condition numbers.

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