Pseudo-Euclidean vector space in the context of Real coordinate space

Pseudo-Euclidean vector space in the context of Real coordinate space

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⭐ Core Definition: Pseudo-Euclidean vector space

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 $(e 1, \dots, e n)$ , be applied to a vector $x = x 1 e 1 + \dots + x n e n$ , giving $q(x)=\left(x_{1}^{2}+\dots +x_{k}^{2}\right)-\left(x_{k+1}^{2}+\dots +x_{n}^{2}\right)$ 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 $1 \leq i \leq k < j \leq n$ , 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.

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⭐ Core Definition: Pseudo-Euclidean vector space
Pseudo-Euclidean vector space in the context of Pseudo-Riemannian manifold

Pseudo-Euclidean vector space in the context of Pseudo-Riemannian manifold

In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.

Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space.

View the full Wikipedia page for Pseudo-Riemannian manifold

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