# Pseudo-Euclidean space

## From Wikipedia, the free encyclopedia

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 (*e*_{1}, …, *e*_{n}), be applied to a vector *x* = *x*_{1}*e*_{1} + ⋯ + *x*_{n}*e*_{n}, 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 ≤ *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).