Dot product

{wiki}

The definition of the "dot product" of a general space varies quite a lot with different contexts.

Most definitions tend to be <bilinear forms>.

We use the unqualified generally refers to the dot product of <Real coordinate spaces>, which is a <positive definite symmetric bilinear form>. Other important examples include:
* the <complex dot product>, which is not strictly <symmetric bilinear map>[symmetric] nor <linear>, but it is <positive definite>
* <Minkowski inner product>, sometimes called" "Minkowski dot product is not <positive definite>
The rest of this section is about the <\R^n> case.

The <positive definite> part of the definition likely comes in because we are so familiar with <metric spaces>, which requires a positive <norm> in the <norm induced by an inner product>.

The default <Euclidean space> definition, we use the <matrix representation of a symmetric bilinear form> as the identity matrix, e.g. in <\R^3>:
$$
M =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
$$
so that:
$$
\vec{x} \cdot \vec{y}
=
\begin{bmatrix}
x_1 & x_2 & x_3 \\
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
y_1 \\
y_2 \\
y_3 \\
\end{bmatrix}
=
x_1y_1 + x_2y_2 + x_3y_3
$$
