The Outer Product

The second type of vector product, the outer product, is less common. If $x$ is a vector of length $n$ and $y$ is a vector of length $m$, then the outer product is denoted as $xy^T$. Again, examine this product in more detail.

$$\begin{eqnarray*} xy^T &=& (n \times 1) * (1 \times m). \\ & & \hspace{.3in} \uparrow \hspace{.25in} \uparrow \end{eqnarray*}$$

The inner dimensions ($1 \times 1$) match. The remaining dimensions are $n \times m,$ hence, an outer product results in a matrix. Also, note that the length of the vectors $x$ and $y$ does not have to be the same.

Let $A = xy^T.$ Then the elements of the outer product matrix $A$ are denoted by

$$a_{i,j} = x_i\,y_j.$$

Finally, as with the dot product, if the vectors are complex then the outer product is defined as $xy^H.$

For $x$ and $y$ defined as above,

$$xy^T = \left(\begin{array}{rrr} 6 & 2 &-6 \\ 0 & 0 & 0 \\ -2 & -1 & 2 \end{array}\right).$$

There are two pictures that are helpful to keep in mind when forming dot and outer products.

$$\begin{array}{lllll} \left(\begin{array}{rrrr} & & & \end{arr... ... & \end{array}\right)&=& \mbox{outer product.} \\ \end{array}$$