Summary
In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0. In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space (X, q) which has a null vector is called a pseudo-Euclidean space. A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, X = A + B, where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres: The null cone is also the union of the isotropic lines through the origin. A composition algebra with a null vector is a split algebra. In a composition algebra (A, +, ×, ), the quadratic form is q(x) = x x. When x is a null vector then there is no multiplicative inverse for x, and since x ≠ 0, A is not a division algebra. In the Cayley-Dickson construction, the split algebras arise in the series bicomplex numbers, biquaternions, and bioctonions, which uses the complex number field as the foundation of this doubling construction due to L. E. Dickson (1919). In particular, these algebras have two imaginary units, which commute so their product, when squared, yields +1: Then so 1 + hi is a null vector. The real subalgebras, split complex numbers, split quaternions, and split-octonions, with their null cones representing the light tracking into and out of 0 ∈ A, suggest spacetime topology. The light-like vectors of Minkowski space are null vectors. The four linearly independent biquaternions l = 1 + hi, n = 1 + hj, m = 1 + hk, and m∗ = 1 – hk are null vectors and { l, n, m, m∗ } can serve as a basis for the subspace used to represent spacetime. Null vectors are also used in the Newman–Penrose formalism approach to spacetime manifolds. In the Verma module of a Lie algebra there are null vectors.
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Related concepts (14)
Isotropic quadratic form
In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More explicitly, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that (V, q) is quadratic space and W is a subspace of V.
Split-quaternion
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers. After introduction in the 20th century of coordinate-free definitions of rings and algebras, it was proved that the algebra of split-quaternions is isomorphic to the ring of the 2×2 real matrices.
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, ..., en), be applied to a vector x = x1e1 + ⋯ + xnen, giving 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, q is an isotropic quadratic form, otherwise it is anisotropic.
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