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 it...

when all its variables are simultaneously zero; otherwise it is isotropic. Quadratic forms occupy a central place in various branches of mathematics, including...

vectors of V. An indefinite quadratic form takes on both positive and negative values and is called an isotropic quadratic form. More generally, these definitions...

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...

is an isotropic quadratic form. If Q has the same sign for all non-zero vectors, it is a definite quadratic form or an anisotropic quadratic form. There...

In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any...

{x^{*}}{N(x)}}} . When there is a non-zero null vector, N is an isotropic quadratic form, and "the algebra splits". Every unital composition algebra over...

z ∗ = x 2 − y 2 , {\displaystyle N(z):=zz^{*}=x^{2}-y^{2},} an isotropic quadratic form. The collection D of all split-complex numbers z = x + y j {\displaystyle...

homogeneity. Isotropic quadratic form A quadratic form q is said to be isotropic if there is a non-zero vector v such that q(v) = 0; such a v is an isotropic vector...

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 ≤ i ≤ k < j ≤...

Minkowski space is a pseudo-Euclidean space equipped with an isotropic quadratic form called the spacetime interval or the Minkowski norm squared. An...

those cases the norm is a definite quadratic form. In the split algebras the norm is an isotropic quadratic form. For any norm p : X → R {\displaystyle...

that a quadratic space over a number field is isotropic if and only if it is isotropic locally everywhere, or equivalently, that a quadratic form over a...

structure of quadratic forms over the field. The universal invariant u(F) of a field F is the largest dimension of an anisotropic quadratic space over F...

an isotropic quadratic form; transformations preserving this form are sometimes called "isometries", and the collection of them is then said to form an...

functions in the x and y over F. Isotropic quadratic forms are multiplicative. For anisotropic quadratic forms, Pfister forms are multiplicative, and conversely...

nondegenerate quadratic forms on vector spaces, the finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely...

~uu^{*}-vv^{*}~} in the definition of SU(1,1) is an Hermitian form which becomes an isotropic quadratic form when u and v are expanded with their real components...

basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field k may be extended...

octonions). There are also composition algebras whose norm is an isotropic quadratic form, which are obtained through a slight modification, by replacing...

identity matrix. isotropic vector In a vector space with a quadratic form, a non-zero vector for which the form is zero. isotropic quadratic form A vector space...

preserve the quadratic form of spacetime and are akin to orthogonal transformations, though with respect to an isotropic quadratic form. The liberties...

n\\0&{\text{otherwise}}\end{cases}}} In this case the symplectic form reduces to a simple quadratic form. If I n {\displaystyle I_{n}} denotes the n × n {\displaystyle...

} and a quadratic form N , {\displaystyle N,} which is called the "norm". In several cases N {\displaystyle N} is an isotropic quadratic form so that...

In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician Cahit Arf (1941)...

Every global and local field is linked since all quadratic forms of degree 6 over such fields are isotropic. The following properties of F are equivalent:: 342 ...

entries. The product of a split-quaternion with its conjugate is the isotropic quadratic form: N ( q ) = q q ∗ = w 2 + x 2 − y 2 − z 2 , {\displaystyle...

This quadratic form N(x) is an isotropic quadratic form since there are non-zero split-octonions x with N(x) = 0. With N, the split-octonions form a pseudo-Euclidean...

119–171; Addendum: ibid, 71(1990); with A.Borel. [7]. Values of isotropic quadratic forms at S-integral points, Compositio Mathematica, 83 (1992), 347–372;...

subspace of its complexification VC = V ⊗R C, and the quadratic form Q extends naturally to a quadratic form QC on VC. This embeds SO(V, Q) as a subgroup of...