Vector Multiplication: Outer Product

The outer product of two vectors is a matrix. If the two vectors have dimensions n and m, then their outer product is an n × m matrix. 

Given two vectors of size \( m\times 1\) and \( n\times 1\)respectively

\[\mathbf {u} ={\begin{bmatrix}u_{1}\\u_{2}\\\vdots \\u_{m}\end{bmatrix}},\quad \mathbf {v} ={\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}\]

their outer product, denoted \( \mathbf {u} \otimes \mathbf {v}\) , is defined as the \( m\times n\) matrix \( \mathbf {A}\)  obtained by multiplying each element of \( \mathbf {u} \)  by each element of \(\mathbf {v}\):

\[\mathbf {u} \otimes \mathbf {v} =\mathbf {A} ={\begin{bmatrix}u_{1}v_{1}&u_{1}v_{2}&\dots &u_{1}v_{n}\\u_{2}v_{1}&u_{2}v_{2}&\dots &u_{2}v_{n}\\\vdots &\vdots &\ddots &\vdots \\u_{m}v_{1}&u_{m}v_{2}&\dots &u_{m}v_{n}\end{bmatrix}}\]

Or in index notation:

\[(\mathbf {u} \otimes \mathbf {v} )_{ij}=u_{i}v_{j}\]

The outer product \(\mathbf {u} \otimes \mathbf {v}\)  is equivalent to a matrix multiplication \(\mathbf {u} \mathbf {v} ^{\operatorname {T} }\), provided that \(\displaystyle \mathbf {u} \)  is represented as a \( m\times 1\) column vector and \( \mathbf {v} \)  as a \( n\times 1\) column vector (which makes \( \mathbf {v} ^{\operatorname {T} }\) a row vector).

For instance, if \( m=4\) and \( n=3,\) then

\[ \mathbf {u} \otimes \mathbf {v} =\mathbf {u} \mathbf {v} ^{\textsf {T}}={\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\\u_{4}\end{bmatrix}}{\begin{bmatrix}v_{1}&v_{2}&v_{3}\end{bmatrix}}={\begin{bmatrix}u_{1}v_{1}&u_{1}v_{2}&u_{1}v_{3}\\u_{2}v_{1}&u_{2}v_{2}&u_{2}v_{3}\\u_{3}v_{1}&u_{3}v_{2}&u_{3}v_{3}\\u_{4}v_{1}&u_{4}v_{2}&u_{4}v_{3}\end{bmatrix}}\]

Useful References

Outer product - Wikipedia