• Journal of applied mathematics & informatics
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• 제14권1_2호
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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.

• 대한수학회지
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• 제54권6호
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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.

• 대한화학회지
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• 제16권6호
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• pp.349-353
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• 1972

다체(4. 이상)계의 Faddeev형 방정식을 Weinberg방정식으로 부터 다른 저자들에 의한 종전의 방법보다 훨씬 간단하게 유도하였다. 유도된 Faddeev형 방정식을 matrix로 표현하였고 matrix적분방정식의 matrix kernel과 inhomogeneous term을 구성하는 방법을 규칙화 하였다. 3,4,5 체계를 예로 들어서 얻어진 규칙들을 실증하였다.

• Current Optics and Photonics
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• 제5권3호
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• pp.322-328
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• 2021

In this paper, a pretreatment method for a matrix in convex optimization is proposed to optimize the spectral reconstruction process of a disordered dispersion spectrometer. Unlike the reconstruction process of traditional spectrometers using Fourier transforms, the reconstruction process of disordered dispersion spectrometers involves solving a large-scale matrix equation. However, since the matrices in the matrix equation are obtained through measurement, they contain uncertainties due to out of band signals, background noise, rounding errors, temperature variations and so on. It is difficult to solve such a matrix equation by using ordinary nonstationary iterative methods, owing to instability problems. Although the smoothing Tikhonov regularization approach has the ability to approximatively solve the matrix equation and reconstruct most simple spectral shapes, it still suffers the limitations of reconstructing complex and irregular spectral shapes that are commonly used to distinguish different elements of detected targets with mixed substances by characteristic spectral peaks. Therefore, we propose a special pretreatment method for a matrix in convex optimization, which has been proved to be useful for reducing the condition number of matrices in the equation. In comparison with the reconstructed spectra gotten by the previous ordinary iterative method, the spectra obtained by the pretreatment method show obvious accuracy.

• Korean Journal of Mathematics
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• 제22권1호
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• pp.123-131
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• 2014

The purpose of this paper is to study the perturbation analysis of the matrix equation $X=I-A^*X^{-1}A+B^*X^{-1}B$. Based on the matrix differentiation, we give a precise perturbation bound for the positive definite solution. A numerical example is presented to illustrate the shrpness of the perturbation bound.

• Structural Engineering and Mechanics
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• 제20권1호
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• pp.111-121
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• 2005

The dynamic interaction of vehicle-bridge is studied by using transfer matrix method in this paper. The vehicle model is simplified as a spring-damping-mass system. By adopting the idea of Newmark-${\beta}$ method, the partial differential equation of structure vibration is transformed into a differential equation irrelevant to time. Then, this differential equation is solved by transfer matrix method. The prospective application of this method in real engineering is finally demonstrated by several examples.

• 대한수학회보
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• 제58권3호
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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.

• 제어로봇시스템학회논문지
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• 제19권9호
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• pp.827-833
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• 2013

In this paper, an efficient computational method is presented for state space analysis of bilinear systems via Legendre wavelets. The differential matrix equation is converted to a generalized Sylvester matrix equation by using Legendre wavelets as a basis. First, an explicit expression for the inverse of the integral operational matrix of the Legendre wavelets is presented. Then using it, we propose a preorder traversal algorithm to solve the generalized Sylvester matrix equation, which greatly reduces the computation time. Finally the efficiency of the proposed method is discussed using numerical examples.

• Journal of applied mathematics & informatics
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• 제35권1_2호
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• pp.165-180
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• 2017

In this study, Taylor matrix algorithm is designed for the approximate solution of linear and non-linear differential equation systems. The algorithm is essentially based on the expansion of the functions in differential equation systems to Taylor series and substituting the matrix forms of these expansions into the given equation systems. Using the Mathematica program, the matrix equations are solved and the unknown Taylor coefficients are found approximately. The presented numerical approach is discussed on samples from various linear and non-linear differential equation systems as well as stiff systems. The computational data are then compared with those of some earlier numerical or exact results. As a result, this comparison demonstrates that the proposed method is accurate and reliable.

• Journal of applied mathematics & informatics
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• 제27권3_4호
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• pp.673-683
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• 2009

We point out: to make Hermtian matrices A and B satisfy Styan matrix inequality, the condition "positive definite property" demanded in the present literatures is not necessary. Furthermore, on the premise of abandoning positive definite property, we derive Styan matrix inequality of Hadamard product for inverse Hermitian matrices and the sufficient and necessary conditions that the equation holds in our paper.