Definition:Outer Product

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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.