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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 Commutation matrix ： ウィキペディア英語版 Commutation matrix In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K(m,n) is the ''mn × mn'' matrix which, for any ''m × n'' matrix A, transforms vec(A) into vec(AT): :K(m,n) vec(A) = vec(AT) . Here vec(A) is the ''mn × 1'' column vector obtain by stacking the columns of A on top of one another: :vec(A) = (A1,1, ..., Am,1, A1,2, ..., Am,2, ..., A1,n, ..., Am,n )T where A = (). The commutation matrix is a special type of permutation matrix, and is therefore orthogonal. Replacing A with AT in the definition of the commutation matrix shows that K(m,n) = (K(n,m))T. Therefore in the special case of m = n the commutation matrix is an involution and symmetric. The main use of the commutation matrix, and the source of its name, is to commute the Kronecker product: for every ''m × n'' matrix A and every ''r × q'' matrix B, :K(r,m)(A $\backslash otimes$ B)K(n,q) = B $\backslash otimes$ A. An explicit form for the commutation matrix is as follows: if er,j denotes the j-th canonical vector of dimension ''r'' (i.e. the vector with 1 in the j-th coordinate and 0 elsewhere) then :K(r,m) = $\backslash sum\_^$$\backslash sum\_^$er,iem,jT$\backslash otimes$em,jer,iT. ==Example== Let ''M'' be a 2x2 square matrix. $$ \mathbf = \begin a & b \\ c & d \\ \end Then we have $$ vec(\mathbf) = \begin a \\ c \\ b \\ d \\ \end And K(2,2) is the 4x4 square matrix that will transform vec(M) into vec(MT) $$ \begin 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end \cdot \begin a \\ c \\ b \\ d \\ \end = \begin a \\ b \\ c \\ d \\ \end = vec(\mathbf^T) 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「Commutation 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.