Pseudo-Euclidean space

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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, …, en), be applied to a vector x = x1e1 + ⋯ + xnen, giving

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, q is an isotropic quadratic form, otherwise it is anisotropic. Note that if 1 ≤ ik < jn, then q(ei + ej) = 0, so that ei + ej 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).