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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 Generalized eigenvector ： ウィキペディア英語版 Generalized eigenvector In linear algebra, a generalized eigenvector of an ''n'' × ''n'' matrix $A$ is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let $V$ be an ''n''-dimensional vector space; let $\backslash phi$ be a linear map in , the set of all linear maps from $V$ into itself; and let $A$ be the matrix representation of $\backslash phi$ with respect to some ordered basis. There may not always exist a full set of ''n'' linearly independent eigenvectors of $A$ that form a complete basis for $V$. That is, the matrix $A$ may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue $\backslash lambda\_i$ is greater than its geometric multiplicity (the nullity of the matrix $(A-\backslash lambda\_i\; I)$, or the dimension of its nullspace). In this case, $\backslash lambda\_i$ is called a defective eigenvalue and $A$ is called a defective matrix. A generalized eigenvector $x\_i$ corresponding to $\backslash lambda\_i$, together with the matrix $(A-\backslash lambda\_i\; I)$ generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of $V$. Using generalized eigenvectors, a set of linearly independent eigenvectors of $A$ can be extended, if necessary, to a complete basis for $V$. This basis can be used to determine an "almost diagonal matrix" $J$ in Jordan normal form, similar to $A$, which is useful in computing certain matrix functions of $A$. The matrix $J$ is also useful in solving the system of linear differential equations $\backslash bold\; x\text{'}\; =\; A\; \backslash bold\; x,$ where $A$ need not be diagonalizable. == Overview and definition == There are several equivalent ways to define an ordinary eigenvector. For our purposes, an eigenvector $\backslash bold\; u$ associated with an eigenvalue $\backslash lambda$ of an $n$ × $n$ matrix $A$ is a nonzero vector for which $(A\; -\; \backslash lambda\; I)\; \backslash bold\; u\; =\; \backslash bold\; 0$, where $I$ is the $n$ × $n$ identity matrix and $\backslash bold\; 0$ is the zero vector of length $n$. That is, $\backslash bold\; u$ is in the kernel of the transformation $(A\; -\; \backslash lambda\; I)$. If $A$ has $n$ linearly independent eigenvectors, then $A$ is similar to a diagonal matrix $D$. That is, there exists an invertible matrix $M$ such that $A$ is diagonalizable through the similarity transformation $D\; =\; M^AM$. The matrix $D$ is called a spectral matrix for $A$. The matrix $M$ is called a modal matrix for $A$. Diagonalizable matrices are of particular interest since matrix functions of them can be computed easily. On the other hand, if $A$ does not have $n$ linearly independent eigenvectors associated with it, then $A$ is not diagonalizable. Definition: A vector $\backslash bold\; x\_m$ is a generalized eigenvector of rank ''m'' of the matrix $A$ and corresponding to the eigenvalue $\backslash lambda$ if :$(A\; -\; \backslash lambda\; I)^m\; \backslash bold\; x\_m\; =\; \backslash bold\; 0$ but :$(A\; -\; \backslash lambda\; I)^\; \backslash bold\; x\_m\; \backslash ne\; \backslash bold\; 0.$ Clearly, a generalized eigenvector of rank 1 is an ordinary eigenvector. Every $n$ × $n$ matrix $A$ has $n$ linearly independent generalized eigenvectors associated with it and can be shown to be similar to an "almost diagonal" matrix $J$ in Jordan normal form. That is, there exists an invertible matrix $M$ such that $J\; =\; M^AM$. The matrix $M$ in this case is called a generalized modal matrix for $A$. If $\backslash lambda$ is an eigenvalue of algebraic multiplicity $\backslash mu$, then $A$ will have $\backslash mu$ linearly independent generalized eigenvectors corresponding to $\backslash lambda$. These results, in turn, provide a straightforward method for computing certain matrix functions of $A$. The set spanned by all generalized eigenvectors for a given $\backslash lambda$, forms the generalized eigenspace for $\backslash lambda$. 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「Generalized eigenvector」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.