并矢积 分量 矩阵表示 参见 导航菜单

张量二元運算


数学双线性代数向量张量积阶张量基列向量乘以行向量矩阵基克罗内克积




在数学特别是双线性代数中，有同样维度的两个向量 udisplaystyle mathbf u 和 vdisplaystyle mathbf v 的并矢积


P=u⊗vdisplaystyle mathbb P =mathbf u otimes mathbf v

是这些向量的张量积，而结果是阶为 2 的张量。



分量


关于选定的基 eidisplaystyle mathbf e _i，并矢积 P=u⊗vdisplaystyle mathbb P =mathbf u otimes mathbf v 的分量 Pijdisplaystyle P_ij 可以定义为



Pij=uivjdisplaystyle P_ij=u_iv_j ,

这里的



u=∑iuieidisplaystyle mathbf u =sum _iu_imathbf e _i ,


v=∑jvjejdisplaystyle mathbf v =sum _jv_jmathbf e _j ,

而



P=∑i,jPijei⊗ejdisplaystyle mathbb P =sum _i,jP_ijmathbf e _iotimes mathbf e _j .


矩阵表示


并矢积可以简单的表示为通过列向量 udisplaystyle mathbf u 乘以行向量 vdisplaystyle mathbf v 的方块矩阵。例如，


u⊗v→[u1u2u3][v1v2v3]=[u1v1u1v2u1v3u2v1u2v2u2v3u3v1u3v2u3v3],displaystyle mathbf u otimes mathbf v rightarrow beginbmatrixu_1\u_2\u_3endbmatrixbeginbmatrixv_1&v_2&v_3endbmatrix=beginbmatrixu_1v_1&u_1v_2&u_1v_3\u_2v_1&u_2v_2&u_2v_3\u_3v_1&u_3v_2&u_3v_3endbmatrix,

这里的箭头指示这只是并矢积关于特定基的特定表示。在这种表示中，并矢积是克罗内克积的特殊情况。



参见


  • 并矢张量

  • 张量积

  • 克罗内克积

  • 外积


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