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 AT . For example, if

A = [ ] 1 2 3 4 5 6
then the transpose of A is
AT = ⌊ 1 4 ⌋ ⌈ ⌉ 2 5 3 6.

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

x = ⌊ ⌋ x1 | | || x2 || | x3 | |⌈ ... |⌉ xn,
the transpose xT is a 1 × n row matrix
xT = [ x x x ⋅⋅⋅ x ] 1 2 3 n.

The dot product as matrix multiplication

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

xT y = [ ] x1 x2 x3 ⋅⋅⋅ xn⌊ ⌋ y1 || y2 || | y | || 3. || ⌈ .. ⌉ yn = x1y1 + x2y2 + x3y3 + … + xnyn = x ⋅ y.
Although we won’t typically write a dot product as xT y, you may see it elsewhere. Moreover, this dot product forms the building block for the general matrix multiplication.