• Title/Summary/Keyword: positive definite matrix

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THE GENERAL HERMITIAN NONNEGATIVE-DEFINITE AND POSITIVE-DEFINITE SOLUTIONS TO THE MATRIX EQUATION $GXG^*\;+\;HYH^*\;=\;C$

  • Zhang, Xian
    • Journal of applied mathematics & informatics
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    • v.14 no.1_2
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    • pp.51-67
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    • 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.

HERMITIAN POSITIVE DEFINITE SOLUTIONS OF THE MATRIX EQUATION Xs + A*X-tA = Q

  • Masoudi, Mohsen;Moghadam, Mahmoud Mohseni;Salemi, Abbas
    • Journal of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.1667-1682
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    • 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.

ON SOME MATRIX INEQUALITIES

  • Lee, Hyun Deok
    • Korean Journal of Mathematics
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    • v.16 no.4
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    • pp.565-571
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    • 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.

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ON POSITIVE DEFINITE SOLUTIONS OF A CLASS OF NONLINEAR MATRIX EQUATION

  • Fang, Liang;Liu, San-Yang;Yin, Xiao-Yan
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.431-448
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    • 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.

Weighted Carlson Mean of Positive Definite Matrices

  • Lee, Hosoo
    • Kyungpook Mathematical Journal
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    • v.53 no.3
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    • pp.479-495
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    • 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.

CONVERGENCE OF MULTISPLITTING METHOD FOR A SYMMETRIC POSITIVE DEFINITE MATRIX

  • YUN JAE HEON;OH SEYOUNG;KIM EUN HEUI
    • Journal of applied mathematics & informatics
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    • v.18 no.1_2
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    • pp.59-72
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    • 2005
  • We study convergence of symmetric multisplitting method associated with many different multisplittings for solving a linear system whose coefficient matrix is a symmetric positive definite matrix which is not an H-matrix.

MULTI SPLITTING PRECONDITIONERS FOR A SYMMETRIC POSITIVE DEFINITE MATRIX

  • Yun Jae-Heon;Kim Eun-Heui;Oh Se-Young
    • Journal of applied mathematics & informatics
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    • v.22 no.1_2
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    • pp.169-180
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    • 2006
  • We study convergence of multisplitting method associated with a block diagonal conformable multisplitting for solving a linear system whose coefficient matrix is a symmetric positive definite matrix which is not an H-matrix. Next, we study the validity of m-step multisplitting polynomial preconditioners which will be used in the preconditioned conjugate gradient method.

POSITIVENESS FOR THE RIEMANNIAN GEODESIC BLOCK MATRIX

  • Hwang, Jinmi;Kim, Sejong
    • Communications of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.917-925
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    • 2020
  • It has been shown that the geometric mean A#B of positive definite Hermitian matrices A and B is the maximal element X of Hermitian matrices such that $$\(\array{A&X\\X&B}\)$$ is positive semi-definite. As an extension of this result for the 2 × 2 block matrix, we consider in this article the block matrix [[A#wijB]] whose (i, j) block is given by the Riemannian geodesics of positive definite Hermitian matrices A and B, where wij ∈ ℝ for all 1 ≤ i, j ≤ m. Under certain assumption of the Loewner order for A and B, we establish the equivalent condition for the parameter matrix ω = [wij] such that the block matrix [[A#wijB]] is positive semi-definite.

THE EQUIVALENT FORM OF A MATRIX INEQUALITY AND ITS APPLICATION

  • ZHONGPENG YANG;XIAOXIA FENG
    • Journal of applied mathematics & informatics
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    • v.20 no.1_2
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    • pp.421-431
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    • 2006
  • In this paper, we establish a matrix inequality and its equivalent form. Applying the results, some matrix inequalities involving Khatri-Rao products of positive semi-definite matrices are generalized.

TWO INEQUALITIES INVOLVING HADAMARD PRODUCTS OF POSITIVE SEMI-DEFINITE HERMITIAN MATRICES

  • Cao, Chong-Guang;Yang, Zhong-Peng;Xian Zhang
    • Journal of applied mathematics & informatics
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    • v.10 no.1_2
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    • pp.101-109
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    • 2002
  • We extend two inequalities involving Hadamard Products of Positive definite Hermitian matrices to positive semi-definite Hermitian matrices. Simultaneously, we also show the sufficient conditions for equalities to hold. Moreover, some other matrix inequalities are also obtained. Our results and methods we different from those which are obtained by S. Liu in [J. Math. Anal. Appl. 243:458-463(2000)] and B.-Y Wang et al in [Lin. Alg. Appl. 302-303: 163-172(1999)] .