翻訳と辞書 Words near each other ・ Jordan Malloch ・ Jordan Malone ・ Jordan Manihera ・ Jordan map ・ Jordan Maritime Authority ・ Jordan Maron ・ Jordan Marsh ・ Jordan Marsh Co. v. Commissioner ・ Jordan Marshall ・ Jordan Marshall (footballer born 1996) ・ Jordan Martinez ・ Jordan Martinook ・ Jordan Massengo ・ Jordan Masterson ・ Jordan Matechuk ・ Jordan matrix ・ Jordan Mattern ・ Jordan Matthews ・ Jordan McCloskey ・ Jordan McColl ・ Jordan McCoy ・ Jordan McDeere ・ Jordan McGhee ・ Jordan McGrotty ・ Jordan McIntosh ・ Jordan McKechnie ・ Jordan McLean ・ Jordan McLean (musician) ・ Jordan McMahon ・ Jordan McMillan Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Jordan matrix ： ウィキペディア英語版 Jordan matrix In the mathematical discipline of matrix theory, a Jordan block over a ring $R$ (whose identities are the zero 0 and one 1) is a matrix composed of 0 elements everywhere except for the diagonal, which is filled with a fixed element $\backslash lambda\backslash in\; R$, and for the superdiagonal, which is composed of ones. The concept is named after Camille Jordan. :$\backslash begin$ \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots& \ddots & \vdots \\ 0 & 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & 0 & \lambda \end Every Jordan block is thus specified by its dimension ''n'' and its eigenvalue $\backslash lambda$ and is indicated as $J\_$. Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix; using either the $\backslash oplus$ or the “$\backslash mathrm$” symbol, the $(l+m+n)\backslash times\; (l+m+n)$ block diagonal square matrix whose first diagonal block is $J\_$, whose second diagonal block is $J\_$ and whose third diagonal block is $J\_$ is compactly indicated as $J\_\backslash oplus\; J\_\backslash oplus\; J\_$ or $\backslash mathrm\backslash left(J\_,\; J\_,\; J\_\backslash right)$, respectively. For example the matrix :$$ J=\left(\begin 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & i & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & i & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & i & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 \end\right) is a $10\backslash times\; 10$ Jordan matrix with a $3\backslash times\; 3$ block with eigenvalue $0$, two $2\backslash times\; 2$ blocks with eigenvalue the imaginary unit and a $3\backslash times\; 3$ block with eigenvalue 7. Its Jordan-block structure can also be written as either $J\_\backslash oplus\; J\_\backslash oplus\; J\_\backslash oplus\; J\_$ or $\backslash mathrm\backslash left(J\_,J\_,J\_,J\_\backslash right)$. == Linear algebra == Any $n\backslash times\; n$ square matrix $A$ whose elements are in an algebraically closed field $K$ is similar to a Jordan matrix $J$, also in $\backslash mathbb\_n\; (K)$, which is unique up to a permutation of its diagonal blocks themselves. $J$ is called the Jordan normal form of $A$ and corresponds to a generalization of the diagonalization procedure. A diagonalizable matrix is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all $1\backslash times\; 1$. More generally, given a Jordan matrix $J=J\_\backslash oplus\; J\_\; \backslash oplus\backslash ldots\backslash oplus\; J\_$, i.e. whose $k^\backslash text$ diagonal block, $1\backslash leq\; k\backslash leq\; N$ is the Jordan block $J\_$ and whose diagonal elements $\backslash lambda\_k$ may not all be distinct, the geometric multiplicity of $\backslash lambda\backslash in\; K$ for the matrix $J$, indicated as $\backslash mathrm\_J\; \backslash lambda\backslash ,$, corresponds to the number of Jordan blocks whose eigenvalue is $\backslash lambda$. Whereas the index of an eigenvalue $\backslash lambda$ for $J$, indicated as $\backslash mathrm\_J\; \backslash lambda\backslash ,$, is defined as the dimension of the largest Jordan block associated to that eigenvalue. The same goes for all the matrices $A$ similar to $J$, so $\backslash mathrm\_A\; \backslash lambda\backslash ,$ can be defined accordingly with respect to the Jordan normal form of $A$ for any of its eigenvalues $\backslash lambda\; \backslash in\backslash mathrmA$. In this case one can check that the index of $\backslash lambda$ for $A$ is equal to its multiplicity as a root of the minimal polynomial of $A$ (whereas, by definition, its algebraic multiplicity for $A$, $\backslash mathrm\_A\; \backslash lambda\backslash ,$, is its multiplicity as a root of the characteristic polynomial of $A$, i.e. $\backslash det(A-xI)\backslash in\; K()$). An equivalent necessary and sufficient condition for $A$ to be diagonalizable in $K$ is that all of its eigenvalues have index equal to $1$, i.e. its minimal polynomial has only simple roots. Note that knowing a matrix's spectrum with all of its algebraic/geometric multiplicities and indexes does not always allow for the computation of its Jordan normal form (this may be a sufficient condition only for spectrally simple, usually low-dimensional matrices): the Jordan decomposition is, in general, a computationally challenging task. From the vector space point of view, the Jordan decomposition is equivalent to finding an orthogonal decomposition (i.e. via direct sums of eigenspaces represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a basis for. 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「Jordan 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.