Dot product in matrix notation

The transpose of a matrix

Before describing the dot product in matrix notation, we briefly digress to make sure you know what the transpose of a matrix is.

We can transpose a matrix by switching its rows with its columns. We denote the transpose of matrix A by A^T. For example, if

$$\left[\vphantom{ \begin{array}{rrr} 1&2&3 4&5&6 \end{array} }\right. \begin{array}{rrr} 1&2&3 4&5&6 \end{array} \left.\vphantom{ \begin{array}{rrr} 1&2&3 4&5&6 \end{array} }\right]$$

then the transpose of A is

$$\left[\vphantom{ \begin{array}{rr} 1&4 2&5 3&6 \end{array} }\right. \begin{array}{rr} 1&4 2&5 3&6 \end{array} \left.\vphantom{ \begin{array}{rr} 1&4 2&5 3&6 \end{array} }\right]$$

Since an n-dimensional vector x is represented by an n x 1 column matrix,

$$\left[\vphantom{ \begin{array}{c} x_1 x_2 x_3 \vdots x_n \end{array} }\right. \begin{array}{c} x_1 x_2 x_3 \vdots x_n \end{array} \left.\vphantom{ \begin{array}{c} x_1 x_2 x_3 \vdots x_n \end{array} }\right]$$

the transpose x^T is a 1 x n row matrix

$$\left[\vphantom{ \begin{array}{ccccc} x_1& x_2& x_3& \cdots& x_n \end{array} }\right. \begin{array}{ccccc} x_1& x_2& x_3& \cdots& x_n \end{array} \left.\vphantom{ \begin{array}{ccccc} x_1& x_2& x_3& \cdots& x_n \end{array} }\right]$$

The dot product as matrix multiplication

Given the rules of matrix multiplication, the product of a 1 x n matrix with an n x 1 matrix is a 1 x 1 matrix, i.e., a scalar. So if we multiply x^T (a 1 x n matrix) with any n-dimensional vector y (viewed as an n x 1 matrix), we end up with a matrix multiplication equivalent to the familiar dot product of x ^. y:

$$\left[\vphantom{ \begin{array}{ccccc} x_1& x_2& x_3& \cdots& x_n \end{array} }\right. \begin{array}{ccccc} x_1& x_2& x_3& \cdots& x_n \end{array} \left.\vphantom{ \begin{array}{ccccc} x_1& x_2& x_3& \cdots& x_n \end{array} }\right] \left[\vphantom{ \begin{array}{c} y_1 y_2 y_3 \vdots y_n \end{array} }\right. \begin{array}{c} y_1 y_2 y_3 \vdots y_n \end{array} \left.\vphantom{ \begin{array}{c} y_1 y_2 y_3 \vdots y_n \end{array} }\right]$$

Although we won't typically write a dot product as x^Ty, you may see it elsewhere. Moreover, this dot product forms the building block for the general matrix multiplication.