A pseudo-Euclidean space is a finite-dimensional real vector space together with a non-degenerate indefinite quadratic form. Such a quadratic form can, after a change of coordinates, be written as

q(x) = \left(x_1^2+\cdots + x_k^2\right)-\left(x_{k+1}^2+\cdots + x_n^2\right)

where x=(x_1, \dots, x_n), n is the dimension of the space, and 1\le k < n.

A very important pseudo-Euclidean space is the Minkowski space, for which n=4 and k=3. For true Euclidean spaces one has k=n, so the quadratic form is positive-definite, rather than indefinite.

Another pseudo-Euclidean space is the plane z = x + y j consisting of split-complex numbers, equipped with the quadratic form

\lVert z \rVert = z z^* = z^* z = x^2 - y^2.

The magnitude of a vector x in the space is defined as q(x). In a pseudo-Euclidean space, unlike in a Euclidean space, there exist non-zero vectors with zero magnitude, and also vectors with negative magnitude.

Associated with the quadratic form q is the pseudo-Euclidean inner product

\langle x, y\rangle = \left(x_1y_1+\cdots + x_ky_k\right)-\left(x_{k+1}y_{k+1}+\cdots + x_ny_n\right).

This bilinear form is symmetric, but not positive-definite, so it is not a true inner product.

An interesting property of pseudo-Euclidean space is that it has not only a unit sphere {x : q(x) = 1 }, but also a counter-sphere {x : q(x) = − 1}. The sets are actually generalized hyperboloids; the term sphere is for consistency with the Euclidean space terminology.

See also

• Pseudo-Riemannian manifold

References

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