Positive-definite bilinear form

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a definite quadratic form is a
quadratic form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
over some
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
that has the same
sign A sign is an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at ...
(always positive or always negative) for every nonzero vector of . 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 "always nonnegative" and "always nonpositive", respectively. In other words, it may take on zero values. An indefinite quadratic form takes on both positive and negative values and is called an
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 precisely, if ''q'' is a quadratic form on a vector ...
. More generally, these definitions apply to any vector space over an
ordered fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

Associated symmetric bilinear form

Quadratic forms correspond one-to-one to
symmetric bilinear formA 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 B tha ...
s over the same space.This is true only over a field of
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
other than 2, but here we consider only
ordered fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s, which necessarily have characteristic 0.
A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form and its associated symmetric bilinear form are related by the following equations: :$\begin Q\left(x\right) &= B\left(x, x\right) \\ B\left(x,y\right) &= B\left(y,x\right) = \frac \left(Q\left(x + y\right) - Q\left(x\right) - Q\left(y\right)\right) \end$ The latter formula arises from expanding $Q\left(x+y\right) = B\left(x+y,x+y\right)$.

Examples

As an example, let $V=\mathbb^2$, and consider the quadratic form :$Q\left(x\right)=c_1^2+c_2^2$ where $\in V$ and and are constants. If and , the quadratic form is positive-definite, so ''Q'' evaluates to a positive number whenever $\left(x_1,x_2\right)\neq \left(0,0\right).$ If one of the constants is positive and the other is 0, then is positive semidefinite and always evaluates to either 0 or a positive number. If and , or vice versa, then 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 $\left(x_1,x_2\right)\neq \left(0,0\right).$ And if one of the constants is negative and the other is 0, then 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: :$Q\left(x\right)=c_1^2+c_2^2+2c_3x_1x_2.$ This quadratic form is positive-definite if $c_1>0$ and $c_1c_2-^2>0,$ negative-definite if $c_1<0$ and $c_1c_2-^2>0,$ and indefinite if $c_1c_2-^2<0.$ It is positive or negative semidefinite if $c_1c_2-^2=0,$ with the sign of the semidefiniteness coinciding with the sign of $c_1.$ This bivariate quadratic form appears in the context of
conic section In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s centered on the origin. If the general quadratic form above is equated to 0, the resulting equation is that of an
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

if the quadratic form is positive or negative-definite, a
hyperbola File:Hyperbel-def-ass-e.svg, 300px, Hyperbola (red): features In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defi ...

if it is indefinite, and a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

if $c_1c_2-^2=0.$ The square of the
Euclidean norm Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
in ''n''-dimensional space, the most commonly used measure of distance, is :$x_1^2+\cdots+x_n^2.$ In two dimensions this means that the distance between two points is the square root of the sum of the squared distances along the $x_1$ axis and the $x_2$ axis.

Matrix form

A quadratic form can be written in terms of
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as :$x^\text A x$ where ''x'' is any ''n''×1 Cartesian vector $\left(x_1, \cdots , x_n\right)^\text$ in which not all elements are 0, superscript T denotes a
transpose In linear algebra, the transpose of a matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matr ...

, and ''A'' is an ''n''×''n''
symmetric matrix In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...
. If ''A'' is
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this is equivalent to a non-matrix form containing solely terms involving squared variables; but if ''A'' has any non-zero off-diagonal elements, the non-matrix form will also contain some terms involving products of two different variables. Positive or negative-definiteness or semi-definiteness, or indefiniteness, of this quadratic form is equivalent to the same property of ''A'', which can be checked by considering all
eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

s of ''A'' or by checking the signs of all of its
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s.

Optimization

Definite quadratic forms lend themselves readily to
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problems. Suppose the matrix quadratic form is augmented with linear terms, as :$x^\text A x + 2 b^\text x ,$ where ''b'' is an ''n''×1 vector of constants. The
first-order conditionIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
s for a maximum or minimum are found by setting the matrix derivative to the zero vector: :$2Ax+2b=0,$ giving :$x=-A^b$ assuming ''A'' is nonsingular matrix, nonsingular. If the quadratic form, and hence ''A'', is positive-definite, the second partial derivative test, second-order conditions for a minimum are met at this point. If the quadratic form is negative-definite, the second-order conditions for a maximum are met. An important example of such an optimization arises in multiple regression, in which a vector of estimated parameters is sought which minimizes the sum of squared deviations from a perfect fit within the dataset.