Definition:Outer Product

Definition
Given two vectors $\vec U = \left({u_1, u_2, \ldots, u_m}\right)$ and $\vec V = \left({v_1, v_2, \ldots ,v_n}\right)$, their outer product $\vec U \otimes \vec V$ is defined as:


 * $U \otimes V = A = \begin{bmatrix}

u_1 v_1 & u_1 v_2 & \dots & u_1 v_n \\ u_2 v_1 & u_2 v_2 & \dots & u_2 v_n \\ \vdots & \vdots & \ddots & \vdots \\ u_m v_1 & u_m v_2 & \dots & u_m v_n \end{bmatrix}$

Index Notation
Given two vectors $u_i$ and $v_j$, their outer product $u_i \otimes v_j$ is defined as


 * $u_i \otimes v_j = a_{ij} = u_i v_j$

Matrix Multiplication
Given two vectors expressed as column matrices $\mathbf{u}$ and $\mathbf{v}$, their outer product $\mathbf{u}\otimes\mathbf{v}$ is defined as


 * $\mathbf{u} \otimes \mathbf{v} = A = \mathbf{u} \mathbf{v}^T$

Properties

 * $A \mathbf{v} = \mathbf{u} \left\|{\mathbf{v}}\right\|^2$: see Vector Length.


 * $\mathbf{u} \otimes \mathbf{v} = \left({\mathbf{v} \otimes \mathbf{u}}\right)^T$