• Journal of applied mathematics & informatics
• /
• 제14권1_2호
• /
• pp.51-67
• /
• 2004

A matrix pair $(X_0,\;Y_0)$ is called a Hermitian nonnegative-definite(respectively, positive-definite) solution to the matrix equation $GXG^*\;+\;HYH^*\;=\;C$ with unknown X and Y if $X_{0}$ and $Y_{0}$ are Hermitian nonnegative-definite (respectively, positive-definite) and satisfy $GX_0G^*\;+\;HY_0H^*\;=\;C$. Necessary and sufficient conditions for the existence of at least a Hermitian nonnegative-definite (respectively, positive-definite) solution to the matrix equation are investigated. A representation of the general Hermitian nonnegative-definite (respectively positive-definite) solution to the equation is also obtained when it has such solutions. Two presented examples show these advantages of the proposed approach.

• 대한수학회보
• /
• 제40권4호
• /
• pp.603-611
• /
• 2003

In this paper we extend the definition of K-positive definite operators from linear to Frechet differentiable operators. Under this setting, we derive from the inverse function theorem a local existence and approximation results corresponding to those of Theorems land 2 of the authors [8], in an arbitrary real Banach space. Furthermore, an asymptotically K-positive definite operator is introduced and a simplified iteration sequence which converges to the unique solution of an asymptotically K-positive definite operator equation is constructed.

• 대한수학회지
• /
• 제54권6호
• /
• pp.1667-1682
• /
• 2017

In this paper, the Hermitian positive definite solutions of the matrix equation $X^s+A^*X-^tA=Q$, where Q is an $n{\times}n$ Hermitian positive definite matrix, A is an $n{\times}n$ nonsingular complex matrix and $s,t{\in}[1,{\infty})$ are discussed. We find a matrix interval which contains all the Hermitian positive definite solutions of this equation. Also, a necessary and sufficient condition for the existence of these solutions is presented. Iterative methods for obtaining the maximal and minimal Hermitian positive definite solutions are proposed. The theoretical results are illustrated by numerical examples.

• 대한수학회보
• /
• 제55권2호
• /
• pp.431-448
• /
• 2018

This paper is concerned with the positive definite solutions of the nonlinear matrix equation $X-A^*{\bar{X}}^{-1}A=Q$, where A, Q are given complex matrices with Q positive definite. We show that such a matrix equation always has a unique positive definite solution and if A is nonsingular, it also has a unique negative definite solution. Moreover, based on Sherman-Morrison-Woodbury formula, we derive elegant relationships between solutions of $X-A^*{\bar{X}}^{-1}A=I$ and the well-studied standard nonlinear matrix equation $Y+B^*Y^{-1}B=Q$, where B, Q are uniquely determined by A. Then several effective numerical algorithms for the unique positive definite solution of $X-A^*{\bar{X}}^{-1}A=Q$ with linear or quadratic convergence rate such as inverse-free fixed-point iteration, structure-preserving doubling algorithm, Newton algorithm are proposed. Numerical examples are presented to illustrate the effectiveness of all the theoretical results and the behavior of the considered algorithms.

• Kyungpook Mathematical Journal
• /
• 제53권3호
• /
• pp.479-495
• /
• 2013

Taking the weighted geometric mean [11] on the cone of positive definite matrix, we propose an iterative mean algorithm involving weighted arithmetic and geometric means of $n$-positive definite matrices which is a weighted version of Carlson mean presented by Lee and Lim [13]. We show that each sequence of the weigthed Carlson iterative mean algorithm has a common limit and the common limit of satisfies weighted multidimensional versions of all properties like permutation symmetry, concavity, monotonicity, homogeneity, congruence invariancy, duality, mean inequalities.

• 대한수학회보
• /
• 제38권2호
• /
• pp.231-236
• /
• 2001

In this note we construct a sequence of Picard iterates suitable for the approximation of solutions of K-positive definite operator equations in arbitrary real Banach spaces. Explicit error estimate is obtained and convergence is shown to be as fast as a geometric progression.

• Korean Journal of Mathematics
• /
• 제16권4호
• /
• pp.565-571
• /
• 2008

In this paper we present some trace inequalities for positive definite matrices in statistical mechanics. In order to prove the method of the uniform bound on the generating functional for the semi-classical model, we use some trace inequalities and matrix norms and properties of trace for positive definite matrices.

• 대한수학회보
• /
• 제33권4호
• /
• pp.545-551
• /
• 1996

It is well known in the theory of distributions and proved in [GS, S] that $$ (i) (Bochner-Schwartz) Every positive definite (tempered) distribution is the Fourier transform of a positive tempered measure \mu. $$ $$ (ii) (Schwartz kernel theorem) Let B(\varphi, \psi) be a bilinear distribution. Then for some u \in D'(R^n \times R^n) B(\varphi, \psi) = u(\varphi(x)\bar{\psi}(y)) for every \varphi, \psi \in C_c^\infty. $$ $$ (iii) A translation invariant positive definite bilinear distribution B(\varphi, \psi) is of the form B(\varphi, \psi) = \smallint \varphi(x)\psi(x) d\mu(x) for every \varphi, \psi \in C_c^\infty (R^n), where \mu is a positive tempered measure.

• 대한수학회지
• /
• 제33권3호
• /
• pp.651-655
• /
• 1996

Lagrange's famous Four Square Theorem [L] says that every positive integer can be represented by the sum of four squares. This marvelous theorem was generalized by Mordell [M1] and Ko [K1] as follows : every positive definite integral quadratic form of two, three, four, and five variables is represented by the sum of five, six, seven, and eight squares, respectively. And they tried to extend this to positive definite integral quadratic forms of six or more variables.

• 대한수학회논문집
• /
• 제11권1호
• /
• pp.33-41
• /
• 1996

In this article, all the universal totally positive integral ternary lattices over $Q(\sqrt{5})$ are determined.