# Minkowski space

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.

## Minkowski inner product

This form is not really an inner product in the common modern definition, because it is not positive definite, only a symmetric bilinear form.

## Minkowski inner product matrix ()

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: $$ημν​=⎣⎢⎢⎢⎡​−1000​0100​0010​0001​⎦⎥⎥⎥⎤​ (1)$$ but another equivalent one is to use a time positive representation: $$ημν​=⎣⎢⎢⎢⎡​1000​0−100​00−10​000−1​⎦⎥⎥⎥⎤​ (2)$$ The matrix is typically denoted by the Greek letter eta.