vector - vector product

the inner product

Givens two vectors \(x,y\in \mathbb{R}^n\), the quantity \(x^\top y\), sometimes called the inner product or dot product of the vectors, is a real number given by:
\[x^\top y=\begin{bmatrix}x_1 , x_2 ,\cdots ,x_n \end{bmatrix}\begin{bmatrix}y_1 \\ y_2 \\ \vdots \\ y_n\end{bmatrix}=\sum_{i=1}^n x_iy_i\]
the inner products are reaully just special case of matrix multiplication.

the outer product

Given vectors \(x\in\mathbb{R}^m,y\in \mathbb{R}^n\)(not necessarily of the same size), \(xy^\top\in\mathbb{R}^{m\times n}\) is called the outer product of the vectors, It is a matrix whose entries are given by \((xy^\top)_{ij}=x_i y_i\) ,i.e.,
\[x y^\top=\begin{bmatrix}x_1 \\ x_2 \\\vdots \\x_m \end{bmatrix}\begin{bmatrix}y_1, y_2, \cdots, y_n\end{bmatrix}=\begin{bmatrix}x_1 y_1, x_1 y_2,\cdots x_1 y_n\\
x_2 y_1, x_2 y_2, \cdots, x_2 y_n\\ \cdots, \cdots, \cdots ,\cdots\\
x_m y_1,x_m y_2, \cdots, x_m y_n\end{bmatrix}\]