In mathematics and theoretical physics, a pseudo-Euclidean space of signature (k, n-k) is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e₁, …, e_n), be applied to a vector x = x₁e₁ + ⋯ + x_ne_n, giving $${\displaystyle q(x)=\left(x_{1}^{2}+\dots +x_{k}^{2}\right)-\left(x_{k+1}^{2}+\dots +x_{n}^{2}\right)}$$ which is called the scalar square of the vector x.

For Euclidean spaces, k = n, implying that the quadratic form is positive-definite. When 0 < k < n, then q is an isotropic quadratic form. Note that if 1 ≤ i ≤ k < j ≤ n, then q(e_i + e_j) = 0, so that e_i + e_j is a null vector. In a pseudo-Euclidean space with k < n, unlike in a Euclidean space, there exist vectors with negative scalar square.

As with the term Euclidean space, the term pseudo-Euclidean space may be used to refer to an affine space or a vector space depending on the author, with the latter alternatively being referred to as a pseudo-Euclidean vector space (see point–vector distinction).

The geometry of a pseudo-Euclidean space is consistent despite some properties of Euclidean space not applying, most notably that it is not a metric space as explained below. The affine structure is unchanged, and thus also the concepts line, plane and, generally, of an affine subspace (flat), as well as line segments.

A null vector is a vector for which the quadratic form is zero. Unlike in a Euclidean space, such a vector can be non-zero, in which case it is self-orthogonal. If the quadratic form is indefinite, a pseudo-Euclidean space has a linear cone of null vectors given by { x | q(x) = 0 }. When the pseudo-Euclidean space provides a model for spacetime (see below), the null cone is called the light cone of the origin.

The null cone separates two open sets, respectively for which q(x) > 0 and q(x) < 0. If k ≥ 2, then the set of vectors for which q(x) > 0 is connected. If k = 1, then it consists of two disjoint parts, one with x₁ > 0 and another with x₁ < 0. Similarly, if n − k ≥ 2, then the set of vectors for which q(x) < 0 is connected. If n − k = 1, then it consists of two disjoint parts, one with x_n > 0 and another with x_n < 0.

The quadratic form q corresponds to the square of a vector in the Euclidean case. To define the vector norm (and distance) in an invariant manner, one has to get square roots of scalar squares, which leads to possibly imaginary distances; see square root of negative numbers. But even for a triangle with positive scalar squares of all three sides (whose square roots are real and positive), the triangle inequality does not hold in general.

Hence terms norm and distance are avoided in pseudo-Euclidean geometry, which may be replaced with scalar square and interval respectively.