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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 Eigendecomposition of a matrix ： ウィキペディア英語版 Eigendecomposition of a matrix In the mathematical discipline of linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. ==Fundamental theory of matrix eigenvectors and eigenvalues== (詳細はif and only if it satisfies the linear equation :$\backslash mathbf\; \backslash mathbf\; =\; \backslash lambda\; \backslash mathbf$ where ''λ'' is a scalar, termed the eigenvalue corresponding to v. That is, the eigenvectors are the vectors that the linear transformation A merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigenvalue equation or the eigenvalue problem. This yields an equation for the eigenvalues :$p\backslash left(\backslash lambda\backslash right)\; :=\; \backslash det\backslash left(\backslash mathbf\; -\; \backslash lambda\; \backslash mathbf\backslash right)=\; 0.\; \backslash !\backslash$ We call ''p''(λ) the characteristic polynomial, and the equation, called the characteristic equation, is an ''N''th order polynomial equation in the unknown ''λ''. This equation will have ''N''λ distinct solutions, where 1 ≤ ''N''λ ≤ ''N'' . The set of solutions, that is, the eigenvalues, is called the spectrum of A. We can factor ''p'' as :$p\backslash left(\backslash lambda\backslash right)=\; (\backslash lambda-\backslash lambda\_1)^(\backslash lambda-\backslash lambda\_2)^\backslash cdots(\backslash lambda-\backslash lambda\_k)^\; =\; 0.\; \backslash !\backslash$ The integer ''n''''i'' is termed the algebraic multiplicity of eigenvalue λ''i''. The algebraic multiplicities sum to ''N'': :$\backslash sum\backslash limits\_^\; =N.$ For each eigenvalue, λ''i'', we have a specific eigenvalue equation :$\backslash left(\backslash mathbf\; -\; \backslash lambda\_i\; \backslash mathbf\backslash right)\backslash mathbf\; =\; 0.\; \backslash !\backslash$ There will be 1 ≤ ''m''''i'' ≤ ''n''''i'' linearly independent solutions to each eigenvalue equation. The ''m''''i'' solutions are the eigenvectors associated with the eigenvalue λ''i''. The integer ''m''''i'' is termed the geometric multiplicity of λ''i''. It is important to keep in mind that the algebraic multiplicity ''n''''i'' and geometric multiplicity ''m''''i'' may or may not be equal, but we always have ''m''''i'' ≤ ''n''''i''. The simplest case is of course when ''m''''i'' = ''n''''i'' = 1. The total number of linearly independent eigenvectors, ''N''v, can be calculated by summing the geometric multiplicities :$\backslash sum\backslash limits\_^\; =N\_\{\backslash mathbf\{v\}\}.$ The eigenvectors can be indexed by eigenvalues, ''i.e.'' using a double index, with v''i'',''j'' being the ''j''th eigenvector for the ''i''th eigenvalue. The eigenvectors can also be indexed using the simpler notation of a single index v''k'', with ''k'' = 1, 2, ..., ''N''v. 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「Eigendecomposition of a matrix」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.