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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 Quaternionic matrix ： ウィキペディア英語版 Quaternionic matrix A quaternionic matrix is a matrix whose elements are quaternions. ==Matrix operations== The quaternions form a noncommutative ring, and therefore addition and multiplication can be defined for quaternionic matrices as for matrices over any ring. Addition. The sum of two quaternionic matrices ''A'' and ''B'' is defined in the usual way by element-wise addition: :$(A+B)\_=A\_+B\_.\backslash ,$ Multiplication. The product of two quaternionic matrices ''A'' and ''B'' also follows the usual definition for matrix multiplication. For it to be defined, the number of columns of ''A'' must equal the number of rows of ''B''. Then the entry in the ''i''th row and ''j''th column of the product is the dot product of the ''i''th row of the first matrix with the ''j''th column of the second matrix. Specifically: :$(AB)\_=\backslash sum\_s\; A\_B\_.\backslash ,$ For example, for :$$ U = \begin u_ & u_\\ u_ & u_\\ \end, \quad V = \begin v_ & v_\\ v_ & v_\\ \end, the product is :$$ UV = \begin u_v_+u_v_ & u_v_+u_v_\\ u_v_+u_v_ & u_v_+u_v_\\ \end. Since quaternionic multiplication is noncommutative, care must be taken to preserve the order of the factors when computing the product of matrices. The identity for this multiplication is, as expected, the diagonal matrix I = diag(1, 1, ... , 1). Multiplication follows the usual laws of associativity and distributivity. The trace of a matrix is defined as the sum of the diagonal elements, but in general :$\backslash operatorname(AB)\backslash ne\backslash operatorname(BA).$ Left scalar multiplication is defined by :$(cA)\_=cA\_,\; \backslash qquad\; (Ac)\_=A\_c.\backslash ,$ Again, since multiplication is not commutative some care must be taken in the order of the factors. 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「Quaternionic 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.