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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 shift matrix ： ウィキペディア英語版 shift matrix In mathematics, a shift matrix is a binary matrix with ones only on the superdiagonal or subdiagonal, and zeroes elsewhere. A shift matrix ''U'' with ones on the superdiagonal is an upper shift matrix. The alternative subdiagonal matrix ''L'' is unsurprisingly known as a lower shift matrix. The (''i'',''j''):th component of ''U'' and ''L'' are :$U\_\; =\; \backslash delta\_,\; \backslash quad\; L\_\; =\; \backslash delta\_,$ where $\backslash delta\_$ is the Kronecker delta symbol. For example, the ''5×5'' shift matrices are ::$$ U_5=\begin 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end \quad L_5=\begin 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end. Clearly, the transpose of a lower shift matrix is an upper shift matrix and vice versa. As a linear transformation, a lower shift matrix shifts the components of a row vector one position to the right, with a zero appearing in the first position. An upper shift matrix shifts the components of a row vector one position to the left, with a zero appearing in the last position. Premultiplying a matrix ''A'' by a lower shift matrix results in the elements of ''A'' being shifted downward by one position, with zeroes appearing in the top row. Postmultiplication by a lower shift matrix results in a shift left. Similar operations involving an upper shift matrix result in the opposite shift. Clearly all shift matrices are nilpotent; an ''n'' by ''n'' shift matrix ''S'' becomes the null matrix when raised to the power of its dimension ''n''. ==Properties== Let ''L'' and ''U'' be the ''n'' by ''n'' lower and upper shift matrices, respectively. The following properties hold for both ''U'' and ''L''. Let us therefore only list the properties for ''U'': * det(''U'') = 0 * trace(''U'') = 0 * rank(''U'') = ''n''−1 * The characteristic polynomials of ''U'' is :$p\_U(\backslash lambda)\; =\; (-1)^n\backslash lambda^n.$ * ''U''''n'' = 0. This follows from the previous property by the Cayley–Hamilton theorem. * The permanent of ''U'' is ''0''. The following properties show how ''U'' and ''L'' are related: * ''L''T = ''U''; ''U''T = L *The null spaces of ''U'' and ''L'' are :$N(U)\; =\; \backslash operatorname\backslash ,$ :$N(L)\; =\; \backslash operatorname\backslash .$ * The spectrum of ''U'' and ''L'' is $\backslash$. The algebraic multiplicity of ''0'' is ''n'', and its geometric multiplicity is ''1''. From the expressions for the null spaces, it follows that (up to a scaling) the only eigenvector for ''U'' is $(1,0,\backslash ldots,\; 0)^T$, and the only eigenvector for ''L'' is $(0,\backslash ldots,\; 0,1)^T$. * For ''LU'' and ''UL'' we have :$UL\; =\; I\; -\; \backslash operatorname(0,\backslash ldots,\; 0,1),$ :$LU\; =\; I\; -\; \backslash operatorname(1,0,\backslash ldots,\; 0).$ :These matrices are both idempotent, symmetric, and have the same rank as ''U'' and ''L'' * ''L''''n-a''''U''''n-a'' + ''L''''a''''U''''a'' = ''U''''n-a''''L''''n-a'' + ''U''''a''''L''''a'' = ''I'' (the identity matrix), for any integer ''a'' between 0 and ''n'' inclusive. If ''N'' is any nilpotent matrix, then ''N'' is similar to a block diagonal matrix of the form :$\backslash begin$ S_1 & 0 & \ldots & 0 \\ 0 & S_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & S_r \end where each of the blocks ''S''1, ''S''2, ..., ''S''''r'' is a shift matrix (possibly of different sizes). 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「shift 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.