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In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite .
The quadratic form q is positive definite if q(v) > 0 (similarly, negative definite if q(v) < 0) for every nonzero vector v. [6] When q ( v ) assumes both positive and negative values, q is an isotropic quadratic form .
It follows from the above definitions that a matrix is positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form.In other words, a matrix is positive-definite if and only if it defines an inner product.
A quadratic form with integer coefficients is called an integral binary quadratic form, ... is a positive definite quadratic form then ...
In mathematics, the signature (v, p, r) [clarification needed] of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix g ab of the metric tensor with respect to a basis.
For any bilinear form B : V × V → K, there exists an associated quadratic form Q : V → K defined by Q : V → K : v ↦ B(v, v). When char(K) ≠ 2, the quadratic form Q is determined by the symmetric part of the bilinear form B and is independent of the antisymmetric part. In this case there is a one-to-one correspondence between the ...
Definite quadratic form; Discriminant; Donaldson's theorem; E. E8 lattice; Ε-quadratic form; ... Quadratic form (statistics) Surgery structure set; Sylvester's law ...
Definite form may refer to: Definite quadratic form in mathematics; Definiteness in linguistics This page was last edited on 28 ...