• Korean Journal of Mathematics
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• v.9 no.1
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• pp.53-60
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• 2001

We study the properties of positivity of matrices and construct useful positive matrices. As an application, we consider a directed graph with matrices such that all the associated matrices related to the positive linear functional are positive.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.839-850
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• 2008

In this paper, the symmetric accelerated overrelaxation (SAOR) method for solving $n{\times}n$ fuzzy linear system is discussed, then the convergence theorems in the special cases where matrix S in augmented system SX = Y is H-matrices or consistently ordered matrices and or symmetric positive definite matrices are also given out. Numerical examples are presented to illustrate the theory.

• 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.

• 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.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.669-677
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• 1996

Let $M_n$ be the $C^*$-algebra of all $n \times n$ matrices over the complex field, and $P[M_m, M_n]$ the convex cone of all positive linear maps from $M_m$ into $M_n$ that is, the maps which send the set of positive semidefinite matrices in $M_m$ into the set of positive semi-definite matrices in $M_n$. The convex structures of $P[M_m, M_n]$ are highly complicated even in low dimensions, and several authors [CL, KK, LW, O, R, S, W]have considered the possibility of decomposition of $P[M_m, M_n] into subcones.

• Communications for Statistical Applications and Methods
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• v.22 no.4
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• pp.321-331
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• 2015

The K-means clustering algorithm is a popular and widely used method for clustering. For covariance matrices, we consider a geodesic clustering algorithm based on the K-means clustering framework in consideration of symmetric positive definite matrices as a Riemannian (non-Euclidean) manifold. This paper considers a geodesic clustering algorithm for data consisting of symmetric positive definite (SPD) matrices, utilizing the Riemannian geometric structure for SPD matrices and the idea of a K-means clustering algorithm. A K-means clustering algorithm is divided into two main steps for which we need a dissimilarity measure between two matrix data points and a way of computing centroids for observations in clusters. In order to use the Riemannian structure, we adopt the geodesic distance and the intrinsic mean for symmetric positive definite matrices. We demonstrate our proposed method through simulations as well as application to real financial data.

• 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)] .

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.609-616
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• 2021

We study the solvability of the Fermat's matrix equation in some classes of 2-by-2 matrices. We prove the Fermat's matrix equation has infinitely many solutions in a set of 2-by-2 positive semidefinite integral matrices, and has no nontrivial solutions in some classes including 2-by-2 symmetric rational matrices and stochastic quadratic field matrices.

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.629-639
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• 1994

Loewner ([13]) showend the semigroup of the real non-singular totally positive matrices is generated by itsinfinitesimal elements, that is,s the set of all Jacobi matrices with non-negative off-diagonal elements. In general, a semigroup is not completely recreated from the set of its infinitesimal elements. We extend Loewner's work and show that the semi-group of all invertible upper (or lower) triangular stochastic matrices is generated by the set of its infinitesimal elements.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.439-448
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• 1994

It is known that the steepest descent(SD) method and the conjugate gradient(CG) method [1, 2, 5, 6] converge when these methods are applied to solve linear systems of the form Ax = b, where A is symmetric and positive definite. For some finite difference discretizations of elliptic problems, one gets positive definite matrices that are almost symmetric. Practically, the SD method and the CG method work for these matrices. However, the convergence of these methods is not guaranteed theoretically. The SD method is also called Orthores(1) in iterative method papers. Elman [4] states that the convergence proof for Orthores($\kappa$), with $\kappa$ a positive integer, is not heard. In this paper, we prove that the SD method and the CG method converge when the $\iota$$^2$ matrix norm of the non-symmetric part of a positive definite matrix is less than some value related to the smallest and the largest eigenvalues of the symmetric part of the given matrix.(omitted)