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Concept
Definite quadratic form
Summary
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.
A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively. In other words, it may take on zero values for some non-zero 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 apply to any vector space over an ordered field.
Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space. A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form Q and its associated symmetric bilinear form B are related by the following equations:
The latter formula arises from expanding
As an example, let , and consider the quadratic form
where and c_1 and c_2 are constants. If and the quadratic form Q is positive-definite, so Q evaluates to a positive number whenever If one of the constants is positive and the other is 0, then Q is positive semidefinite and always evaluates to either 0 or a positive number. If and or vice versa, then Q is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If and the quadratic form is negative-definite and always evaluates to a negative number whenever And if one of the constants is negative and the other is 0, then Q is negative semidefinite and always evaluates to either 0 or a negative number.
In general a quadratic form in two variables will also involve a cross-product term in x_1·x_2:
This quadratic form is positive-definite if and negative-definite if and and indefinite if It is positive or negative semidefinite if with the sign of the semidefiniteness coinciding with the sign of
This bivariate quadratic form appears in the context of conic sections centered on the origin.
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Symmetric bilinear form
In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function that maps every pair of elements of the vector space to the underlying field such that for every and in . They are also referred to more briefly as just symmetric forms when "bilinear" is understood.
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.