翻訳と辞書 Words near each other ・ Vector-field consistency ・ Vector-valued differential form ・ Vector-valued function ・ VectorBase ・ Vectorbeam ・ Vectorcardiography ・ VectorCell ・ VectorDB ・ Vectored I/O ・ Vectored Interrupt ・ Vectorial addition chain ・ Vectorial Mechanics ・ Vectorial synthesis ・ Vectoring nozzle ・ Vectorization ・ Vectorization (mathematics) ・ VectorLinux ・ Vectorman ・ Vectors in gene therapy ・ Vectors in three-dimensional space ・ Vectorscope ・ Vectorwise ・ Vectorworks ・ VectorWorks Architect ・ Vectra ・ Vectra (plastic) ・ Vectra Networks Inc. ・ Vectran ・ Vectren ・ Vectren Dayton Air Show 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 Vectorization (mathematics) ： ウィキペディア英語版 Vectorization (mathematics) In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector. Specifically, the vectorization of an ''m×n'' matrix ''A'', denoted by vec(''A''), is the ''mn'' × 1 column vector obtained by stacking the columns of the matrix ''A'' on top of one another: :$\backslash mathrm(A)\; =\; (\backslash ldots,\; a\_,\; a\_,\; \backslash ldots,\; a\_,\; \backslash ldots,\; a\_,\; \backslash ldots,\; a\_)^T$ Here $a\_$ represents the $(i,j)$-th element of matrix $A$ and the superscript $^T$ denotes the transpose. Vectorization expresses the isomorphism $\backslash mathbf^\; :=\; \backslash mathbf^m\; \backslash otimes\; \backslash mathbf^n\; \backslash cong\; \backslash mathbf^$ between these vector spaces (of matrices and vectors) in coordinates. For example, for the 2×2 matrix $A$ = , the vectorization is $\backslash mathrm(A)\; =\; \backslash begin\; a\; \backslash \backslash \; c\; \backslash \backslash \; b\; \backslash \backslash \; d\; \backslash end$. ==Compatibility with Kronecker products== The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular, :$\backslash mbox(ABC)=(C^\backslash otimes\; A)\backslash mbox(B)$ for matrices ''A'', ''B'', and ''C'' of dimensions ''k×l'', ''l×m'', and ''m×n''. For example, if $\backslash mbox\_A(X)\; =\; AX-XA$ (the adjoint endomorphism of the Lie algebra gl(''n'',C) of all ''n×n'' matrices with complex entries), then $\backslash mbox(\backslash mbox\_A(X))\; =\; (I\_n\backslash otimes\; A\; -\; A^T\; \backslash otimes\; I\_n\; )\; \backslash mbox(X)$, where $I\_n$ is the ''n×n'' identity matrix. There are two other useful formulations: :$\backslash mbox(ABC)=(I\_n\backslash otimes\; AB)\backslash mbox(C)\; =(C^B^\backslash otimes\; I\_k)\backslash mbox(A)$ :$\backslash mbox(AB)=(I\_m\backslash otimes\; A)\backslash mbox(B)\; =(B^\backslash otimes\; I\_k)\backslash mbox(A)$ 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「Vectorization (mathematics)」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.