• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.305-318
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• 2019

A sign pattern is a matrix whose entries belong to the set {+, -, 0}. An n-by-n sign pattern ${\mathcal{A}}$ is said to allow an eventually positive matrix or be potentially eventually positive if there exist at least one real matrix A with the same sign pattern as ${\mathcal{A}}$ and a positive integer $k_0$ such that $A^k>0$ for all $k{\geq}k_0$. Identifying the necessary and sufficient conditions for an n-by-n sign pattern to be potentially eventually positive, and classifying the n-by-n sign patterns that allow an eventually positive matrix are two open problems. In this article, we focus on the potential eventual positivity of broom sign patterns. We identify all the minimal potentially eventually positive broom sign patterns. Consequently, we classify all the potentially eventually positive broom sign patterns.

• 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.

• East Asian mathematical journal
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• v.35 no.3
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• pp.341-349
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• 2019

In this paper, we are concerned with the minimal positive solution to system of the nonlinear matrix equations $A_1X^2+B_1Y +C_1=0$ and $A_2Y^2+B_2X+C_2=0$, where $A_i$ is a positive matrix or a nonnegative irreducible matrix, $C_i$ is a nonnegative matrix and $-B_i$ is a nonsingular M-matrix for i = 1, 2. We apply Newton's method to system and present a modified Newton's iteration which is validated to be efficient in the numerical experiments. We prove that the sequences generated by the modified Newton's iteration converge to the minimal positive solution to system of nonlinear matrix equations.

• Communications of the Korean Mathematical Society
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• v.12 no.2
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• pp.269-273
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• 1997

We show that each element in the semigroup $S_n$ of all $n \times n$ non-singular stochastic totally positive matrices is generated by the infinitesimal elements of $S_n$, which form a cone consisting of all $n \times n$ Jacobi intensity matrices.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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#w_ijB]] whose (i, j) block is given by the Riemannian geodesics of positive definite Hermitian matrices A and B, where w_ij ∈ 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 = [w_ij] such that the block matrix [[A#w_ijB]] is positive semi-definite.