Quadratic form Totally Explained

Everything about Quadratic Form totally explained

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. Quadratic forms are central objects in mathematics, occurring for instance in number theory, Riemannian geometry (as curvature), and Lie theory (via the Killing form).
They are also ubiquitous in physics and chemistry, as the energy of a system, particularly in relation to the L² norm, which leads to the use of Hilbert spaces.

Definition

Quadratic forms in one, two, and three variables are given by:
ť

F(x) = ax^2

F(x,y) = ax^2 + by^2 + cxy

F(x,y,z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz

Away from 2, quadratic forms are equivalent to symmetric bilinear forms (by the polarization identities), but at 2 they're different concepts; this distinction is particularly important for quadratic forms over the integers.
   The term quadratic form is also often used to refer to a quadratic space, which is a pair (V,q) where V is a vector space over a field k, and q:V → k is a quadratic form on V. For example, the distance between two points in three-dimensional Euclidean space is found by taking the square root of a quadratic form involving six variables, the three coordinates of each of the two points.
   A quadratic form in 2 variables is called a binary quadratic form, and these are extensively studied in number theory (particularly in the theory of modular forms), together with their associated quadratic fields.
   Note that general quadratic functions and quadratic equations are not examples of quadratic forms, as they're not always homogeneous: quadratic functions are functions on affine space, while quadratic forms are "functions" on projective space (properly, sections of

mathcal

.
Several points of view mean that twos out has been adopted as the standard convention. Those include:

better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty;

the lattice point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s;

the actual needs for integral quadratic form theory in topology for intersection theory;

the Lie group and algebraic group aspects.

Universal quadratic forms

A quadratic form representing all positive integers is sometimes called universal. Lagrange's four-square theorem shows that

w^2+x^2+y^2+z^2

is universal.
Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15.

Real quadratic forms

Assume

Q

is a quadratic form defined on a real vector space.

It is said to be positive definite (resp. negative definite) if

Q(v)>0

(resp.

Q(v)<0

) for every vector

v
e 0.

If we loosen the strict inequality to ≥ or ≤, the form

Q

is said to be semidefinite.

Q(v)<0

for some

v

and

Q(v)>0

for some other

v

Q

is said to be indefinite. Let

A

be the real symmetric matrix associated with

Q

as described above, so for any column vector

v

it holds that ť

Q(v)=v^T Av.

Then,

Q

is positive (semi)definite, negative (semi)definite, indefinite, if and only if the matrix

A

has the same properties (see positive-definite matrix). Ultimately, these properties can be characterized in terms of the eigenvalues of

A.

Further Information

Get more info on 'Quadratic Form'.

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