Vector Direct Product

Given Vectors ${\bf u}$ and ${\bf v}$, the vector direct product is

$${\bf u}{\bf v}\equiv {\bf u}\otimes{\bf v}^{\rm T},$$

where $\otimes$ is the Matrix Direct Product and ${\bf v}^{\rm T}$ is the matrix Transpose. For $3\times 3$ vectors

$${\bf u}{\bf v} = \left[{\matrix{u_1{\bf v}^{\rm T}\cr u_2{\b... ...v_1 & u_2v_2 & u_2v_3\cr u_3v_1 & u_3v_2 & u_3v_3\cr}}\right].$$

Note that if ${\bf u}=\hat{\bf x}_i$, then $u_j=\delta_{ij}$, where $\delta_{ij}$ is the Kronecker Delta.

See also Matrix Direct Product, Sherman-Morrison Formula, Woodbury Formula