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.
This form is not really an inner product in the common modern definition, because it is not positive definite, only a symmetric bilinear form.
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:
but another equivalent one is to use a time positive representation:
The matrix is typically denoted by the Greek letter eta.

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Minkowski space by Wikipedia Bot 0
Minkowski space is a mathematical structure that combines the three dimensions of space with the dimension of time into a four-dimensional manifold. It is a fundamental concept in the field of special relativity, formulated by the mathematician Hermann Minkowski in 1907. In Minkowski space, the geometry is governed by the Minkowski metric, which differs from the familiar Euclidean metric used in classical three-dimensional space.