# Definition:Outer Product

## Definition

Given two vectors $\mathbf u = \left({u_1, u_2, \ldots, u_m}\right)$ and $\mathbf v = \left({v_1, v_2, \ldots ,v_n}\right)$, their **outer product** $\mathbf u \otimes \mathbf v$ is defined as:

- $\mathbf u \otimes \mathbf v = A = \begin{bmatrix} u_1 v_1 & u_1 v_2 & \dots & u_1 v_n \\ u_2 v_1 & u_2 v_2 & \dots & u_2 v_n \\ \vdots & \vdots & \ddots & \vdots \\ u_m v_1 & u_m v_2 & \dots & u_m v_n \end{bmatrix}$

### 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.

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

### Notes

The outer product is sometimes referred to as the **dyad product** between two vectors.