Definition:Outer Product

From ProofWiki
Jump to navigation Jump to search


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

  • $A \mathbf v = \mathbf u \norm {\mathbf v}^2$: see Vector Length.
  • $\mathbf u \otimes \mathbf v = \paren {\mathbf v \otimes \mathbf u}^T$