3.4b Outer Product

Outer Product

The outer product of two vectors $\vec{x}\in {\mathbb{R}}^n$ and $\vec{y}\in {\mathbb{R}}^m$ is given as the matrix

$$\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]\left[\begin{array}{cccc}{y}_1& y_2& \dots & y_m\end{array}\right]=\left[\begin{array}{cccc}{x}_1{y}_1& x_1{y}_2& \dots & x_1{y}_m\\ x_2{y}_1& x_2{y}_2& \dots & x_2{y}_m\\ ⋮& ⋮& \ddots & ⋮\\ x_n{y}_1& x_n{y}_2& \dots & x_n{y}_m\end{array}\right]\in {\mathbb{R}}^{m\times n}.$$