Pseudo-Euclidean space

In mathematics and theoretical physics, a pseudo-Euclidean space 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.^[1]: 3

For Euclidean spaces, k = n, implying that the quadratic form is positive-definite.^[2] 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^[3] (see point–vector distinction).