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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 Characteristic polynomial ： ウィキペディア英語版 Characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial, which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix as coefficients. The characteristic polynomial of an endomorphism of vector spaces of finite dimension is the characteristic polynomial of the matrix of the endomorphism over any base; it does not depend on the choice of a basis. The characteristic equation is the equation obtained by equating to zero the characteristic polynomial. The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. It is a graph invariant, though it is not complete: the smallest pair of non-isomorphic graphs with the same characteristic polynomial have five nodes.〔(【引用サイトリンク】 Characteristic Polynomial of a Graph - Wolfram MathWorld )〕 ==Motivation== Given a square matrix ''A'', we want to find a polynomial whose zeros are the eigenvalues of ''A''. For a diagonal matrix ''A'', the characteristic polynomial is easy to define: if the diagonal entries are ''a''1, ''a''2, ''a''3, etc. then the characteristic polynomial will be: :$(t-a\_1)(t-a\_2)(t-a\_3)\backslash cdots.\backslash ,$ This works because the diagonal entries are also the eigenvalues of this matrix. For a general matrix ''A'', one can proceed as follows. A scalar ''λ'' is an eigenvalue of ''A'' if and only if there is an eigenvector v ≠ 0 such that :$A\; \backslash mathbf\; =\; \backslash lambda\; \backslash mathbf,\backslash ,$ or :$(\backslash lambda\; I\; -\; A)\backslash mathbf\; =\; 0\backslash ,$ (where ''I'' is the identity matrix). Since v is non-zero, this means that the matrix ''λ'' ''I'' − ''A'' is singular (non-invertible), which in turn means that its determinant is 0. Thus the roots of the function det(''λ'' ''I'' − ''A'') are the eigenvalues of ''A'', and it is clear that this determinant is a polynomial in ''λ''. 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「Characteristic polynomial」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.