Pseudo-Euclidean Space

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate indefinite quadratic form, called the magnitude of a vector. Such a quadratic form can, after a suitable change of coordinates, be written as

where x = (x₁, …, x_n), n is the dimension of the space, and 1 ≤ k < n. For true Euclidean spaces, k = n, implying that the quadratic form is positive-definite rather than indefinite. Otherwise q is an isotropic quadratic form. In a pseudo-Euclidean space, unlike in a Euclidean space, there exist non-zero vectors with zero magnitude, and also vectors with negative magnitude.

As with the term Euclidean space, pseudo-Euclidean space may refer to either an affine space or a vector space (see point–vector distinction) over real numbers.