• Journal of the Korean Mathematical Society
• /
• v.57 no.3
• /
• pp.641-653
• /
• 2020

To extend the well-known extremal characterization of the geometric mean of two n × n positive definite matrices A and B, we solve the following problem: $${\max}\{X:X=X^*,\;\(\array{A&V&X\\V&B&W\\X&W&C}\){\geq}0\}$$. We find an explicit expression of the maximum value with respect to the matrix geometric mean of Schur complements.

• Kyungpook Mathematical Journal
• /
• v.53 no.3
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.2
• /
• pp.521-533
• /
• 2019

In this paper, we define a mean of two positive numbers called the k-golden mean and study some properties of it. Especially, we show that the 2-golden mean refines the harmonic and the geometric means. As an application, we define the k-golden ratio and give some properties of it as an generalization of the golden ratio. Furthermore, we define the matrix k-golden mean of two positive-definite matrices and give some properties of it. This is an improvement of Lim's results [2] for which the matrix golden mean.

• Journal of the Chungcheong Mathematical Society
• /
• v.30 no.1
• /
• pp.21-29
• /
• 2017

We review a partial trace alternatively defined by the composition of positive linear maps, and see how the partial trace acts on the matrix geometric means. We also study the quantum Tsallis relative entropy under partial trace related with the fidelity.

• Korean Journal of Remote Sensing
• /
• v.30 no.6
• /
• pp.769-776
• /
• 2014

It is necessary to detect the feature points existing simultaneously in both images and then find the corresponding relationship between the detected feature points. We propose a new feature detector based on geometric mean of two eigenvalues of gradient matrix which is able to measure the change of pixel intensities. The corner response of the proposed detector is proportional to the geometric mean and also the difference of two eigenvalues in the case of same geometric mean. We analyzed the localization error of the feature detection using aerial image and artificial image with various types of corners. The localization error of the proposed detector was smaller than that of the typical corner detector, Harris detector.

• Kyungpook Mathematical Journal
• /
• v.47 no.1
• /
• pp.133-150
• /
• 2007

In a development of efficient primal-dual interior-points algorithms for self-scaled convex programming problems, one of the important properties of such cones is the existence and uniqueness of "scaling points". In this paper through the identification of scaling points with the notion of "(metric) geometric means" on symmetric cones, we extend several well-known matrix inequalities (the classical L$\ddot{o}$wner-Heinz inequality, Ando inequality, Jensen inequality, Furuta inequality) to symmetric cones. We also develop a theory of spectral geometric means on symmetric cones which has recently appeared in matrix theory and in the linear monotone complementarity problem for domains associated to symmetric cones. We derive Nesterov-Todd inequality using the spectral property of spectral geometric means on symmetric cones.

• Journal of the Korean Statistical Society
• /
• v.22 no.2
• /
• pp.201-208
• /
• 1993

Given the specific mean shift outlier model, the score test for multiple outliers in nonlinear regression is discussed as an alternative to the likelihood ratio test. The geometric interpretation of the score statistic is also presented.

• Structural Engineering and Mechanics
• /
• v.22 no.4
• /
• pp.469-481
• /
• 2006

A new method called fuzzy factor method for the stationary stochastic response analysis of fuzzy truss with global fuzzy structural parameters is presented in this paper. Considering the fuzziness of the structural physical parameters and geometric dimensions simultaneously, the fuzzy correlation function matrix of structural displacement response in time domain is derived by using the fuzzy factor method and the optimization method, the fuzzy mean square values of the structural displacement and stress response in the frequency domain are then developed with the fuzzy factor method. The influences of the fuzziness of structural parameters on the fuzziness of mean square values of the displacement and stress response are inspected via an example and some important conclusions are obtained. Finally, the example is simulated by Monte-Carlo method and the results of the two methods are close, which verified the feasibility of the method given in this paper.

• Communications of the Korean Mathematical Society
• /
• v.35 no.3
• /
• pp.917-925
• /
• 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.

• Journal of the Korean Operations Research and Management Science Society
• /
• v.23 no.4
• /
• pp.131-140
• /
• 1998

The objective of this study is to propose a new procedure to synthesize the multi-experts priorities in AHP. If multi-experts with different expertise are involved in a AHP decision, we need some way to aggregate their opinions. AHP model used to do numerical aggregation by taking only the geometric mean or the weighting geometric mean in past. To aggregate the multi-experts priorities, In this paper. we suggest a way which Decision Maker can exclude outlier matrix from group using the concept of the Compatibility and we Introduce Delphi method to use Compatibility in AHP. A numerical example is shown to illustrate the procedure.