# Definition:Outer Product

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## Definition

Given two vectors $\mathbf u = \tuple {u_1, u_2, \ldots, u_m}$ and $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$, 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

- $\mathbf u \otimes \mathbf v = A = \mathbf u \mathbf v^T$

## Also known as

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

## Also see

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- $A \mathbf v = \mathbf u \norm {\mathbf v}^2$: see Vector Length.

- $\mathbf u \otimes \mathbf v = \paren {\mathbf v \otimes \mathbf u}^T$

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**outer product** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**outer product**