Outer Product

linear-algebra

Definition

Outer Product

The outer product of two vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector.

Example: Let $\mathbf{u}\in {\mathbb{F}}^m$ and $\mathbf{v}\in {\mathbb{F}}^n$ be two vectors.

$$u\otimes v=\left[\begin{array}{cccc}{u}_1{v}_1& u_1{v}_2& \cdots & u_1{v}_n\\ u_2{v}_1& u_2{v}_2& \cdots & u_2{v}_n\\ ⋮& ⋮& \ddots & ⋮\\ u_m{v}_1& u_m{v}_2& \cdots & u_m{v}_n\end{array}\right]$$