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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Matrix differential equation ： ウィキペディア英語版 Matrix differential equation A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives. For example, a simple matrix ordinary differential equation is : $\backslash mathbf(t)\; =\; \backslash mathbf(t)$ where x(''t'') is an ''n''×1 vector of functions of an underlying variable , x'(''t'') is the vector of first derivatives of these functions, and A is an matrix, of which all elements are constants. Note that by using the Cayley-Hamilton theorem and Vandermonde-type matrices, a solution may be given in a simple form.〔 H. Moya-Cessa, F. Soto-Eguibar, DIFFERENTIAL EQUATIONS: AN OPERATIONAL APPROACH, RINTON PRESS, New Jersey, 2011. ISBN 978-1-58949-060-4〕 Below the solution is displayed in terms of Putzer's algorithm.〔E. J. Putzer (1966). "Avoiding the Jordan Canonical Form in the Discussion of Linear Systems with Constant Coefficients", ''The American Mathematical Monthly'', 73, No. 1 (1966) 2-7. (), Paper on JSTOR〕 In the case where A has ''n'' distinct eigenvalues, this differential equation has the following general solution: : $\backslash mathbf(t)\; =\; c\_1\; e^\; \backslash mathbf\_1\; +\; c\_2\; e^\; \backslash mathbf\_2\; +\; \backslash cdots\; +\; c\_n\; e^\; \backslash mathbf\_n$ where ''λ''1, ''λ''2, ..., ''λ''''n'' are the eigenvalues of A; u1, u2, ..., u''n'' are the respective eigenvectors of A and ''c''1, ''c''2, ...., ''c''''n'' are constants. ==Stability and steady state of the matrix system== The matrix equation $\backslash mathbf\text{'}(t)\; =\; \backslash mathbf(t)\; +\; \backslash mathbf$ with ''n''×1 parameter vector b is stable if and only if all eigenvalues of the matrix A have a negative real part. The steady state x * to which it converges if stable is found by setting $\backslash mathbf\text{'}(t)=\backslash mathbf$, yielding $\backslash mathbf^$ * = -\mathbf^\mathbf, assuming A is invertible. Thus the original equation can be written in homogeneous form in terms of deviations from the steady state: . A different way of expressing this (closer to regular usage) is that x * is a particular solution to the non-homogeneous equation, and all solutions are in the form $\backslash mathbf\_h+\backslash mathbf^$ *, with $\backslash mathbf\_h$ a solution to the homogeneous equation (b=0). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Matrix differential equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.