{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:
$$
\vec{x} \cdot \vec{y}
=
\begin{bmatrix}
x_1 & x_2 & x_3 \\
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
y_1 \\
y_2 \\
y_3 \\
\end{bmatrix}
=
x_1y_1 + x_2y_2 + x_3y_3
$$


Dot product

{wiki}

The <group (mathematics)> of all transformations that preserve some <bilinear form>, notable examples:
* <orthogonal group> preserves the <inner product>
* <unitary group> preserves a <Hermitian form>
* <Lorentz group> preserves the <Minkowski inner product>


Isometry group

{c}
{tag=Isometry group}
{wiki}

= $SO(1,3)$
{synonym}
{title2}

<Subgroup> of the <Poincaré group> without translations. Therefore, in those, the spacetime origin is always fixed.

Or in other words, it is as if two observers had their space and time origins at the exact same place. However, their space axes may be rotated, and one may be at a relative speed to the other to create a <Lorentz boost>. Note however that if they are at relative speeds to one another, then their axes will immediately stop being at the same location in the next moment of time, so things are only valid infinitesimally in that case.

This group is made up of matrix multiplication alone, no need to add the offset vector: space rotations and <Lorentz boost> only spin around and bend things around the origin.

One definition: set of all 4x4 matrices that keep the <Minkowski inner product>, mentioned at <Physics from Symmetry by Jakob Schwichtenberg (2015)> page 63. This then implies:
$$
\Lambda ^ T \eta \Lambda = \eta
$$


Lorentz group

{c}
{title2=$\eta_{\mu\nu}$}

The <matrix representation of the symmetric bilinear form> that is the <Minkowski inner product>.

Since that is a <symmetric bilinear form>, the associated <matrix> is a <symmetric matrix>.

By default, we will use the time negative representation unless stated otherwise:
$$
\eta_{\mu\nu} =
\begin{bmatrix}
-1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1 \\
\end{bmatrix}
$$
but another equivalent one is to use a time positive representation:
$$
\eta_{\mu\nu} =
\begin{bmatrix}
 1 &  0 &  0 &  0 \\
 0 & -1 &  0 &  0 \\
 0 &  0 & -1 &  0 \\
 0 &  0 &  0 & -1 \\
\end{bmatrix}
$$
The matrix is typically denoted by the <eta>[Greek letter eta].


Minkowski inner product matrix

{c}
{tag=Pseudometric space}
{wiki}

<\R^4> with a weird <dot product>-like operation called the <Minkowski inner product>.

Because the <Minkowski inner product> product is not <positive definite>, the <norm induced by an inner product> is a <norm>, and the space is not a <metric space> strictly speaking.

The name given to this type of space is a <pseudometric space>.


Ciro Santilli @cirosantilli 34

 Incoming links: Minkowski inner product