Quadratic forms

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Indefinite quadratic form

A quadratic form is indefinite if it is less than 0 for some values and greater than 0 for others. The quadratic form, written in the form , is indefinite if eigenvalues of the matrix are of both signs.

Quadratic form

A quadratic form involving real variables , , ..., associated with the matrix is given by(1)where Einstein summation has been used. Letting be a vector made up of , ..., and the transpose, then(2)equivalent to(3)in inner product notation. A binaryquadratic form is a quadratic form in two variables and has the form(4)It is always possible to express an arbitrary quadratic form(5)in the form(6)where is a symmetric matrix given by(7)Any real quadratic form in variables may be reduced to the diagonal form(8)with by a suitable orthogonal point-transformation. Also, two real quadratic forms are equivalent under the group of linear transformations iff they have the same quadratic form rank and quadratic form signature.

Positive semidefinite quadratic form

A quadratic form is said to be positive semidefinite if it is never . However, unlike a positive definite quadratic form, there may exist a such that the form is zero. The quadratic form, written in the form , is positive semidefinite iff every eigenvalue of is nonnegative.

Hermitian form

A Hermitian form on a vector space over the complex field is a function such that for all and all , 1. . 2. . Here, the bar indicates the complex conjugate. It follows that(1)which can be expressed by saying that is antilinear on the second coordinate. Moreover, for all , , which means that .An example is the dot product of , defined as(2)Every Hermitian form on is associated with an Hermitian matrix such that(3)for all row vectors and of . The matrix associated with the dot product is the identity matrix.More generally, if is a vector space on a field , and is an automorphism such that , and , the notation can be used and a Hermitian form on can be defined by means of the properties (1) and (2).

Positive definite quadratic form

A quadratic form is said to be positive definite if for . A real quadratic form in variables is positive definite iff its canonical form is(1)A binary quadratic form(2)of two real variables is positive definite if it is for any , therefore if and the binary quadratic form discriminant . A binary quadratic form is positive definite if there exist nonzero and such that(3)(Le Lionnais 1983).The positive definite quadratic form(4)is said to be reduced if , , and if or . Under the action of the general linear group , i.e., under the set of linear transformations of coordinates with integer coefficients and determinant , there exists a unique reduced positive definite binary quadratic form equivalent to any given one.There exists a one-to-one correspondence between the set of reduced quadratic forms with fundamental discriminant and the set of classes of fractional ideals of the unique quadratic field with discriminant . Let be a reduced positive definite..

Poincaré separation theorem

Let be a set of orthonormal vectors with , 2, ..., , such that the inner product . Then set(1)so that for any square matrix for which the product is defined, the corresponding quadratic form is(2)Then if(3)for , 2, ..., , it follows that(4)(5)for , 2, ..., and , 1, ..., .

Diagonal quadratic form

If is a diagonal matrix, then(1)is a diagonal quadratic form, and is its associated diagonal symmetric bilinear form.For a general symmetric matrix , a symmetric bilinear form may be diagonalized by a nondegenerate matrix such that is a diagonal form. That is, is a diagonal matrix. Note that may not be an orthogonal matrix.For example, consider(2)Then taking the diagonalizer(3)gives the diagonal matrix(4)

Symmetric bilinear form

A symmetric bilinear form on a vector space is a bilinear function(1)which satisfies .For example, if is a symmetric matrix, then(2)is a symmetric bilinear form. Consider(3)then(4)A quadratic form may also be labeled , because quadratic forms are in a one-to-one correspondence with symmetric bilinear forms. Note that is a quadratic form. If is a quadratic form then it defines a symmetric bilinear form by(5)The kernel, or radical, of a symmetric bilinear form is the set of vectors(6)A quadratic form is called nondegenerate if its kernel is zero. That is, if for all , there is a with . The rank of is the rank of the matrix .The form is diagonalized if there is a basis , called an orthogonal basis, such that is a diagonal matrix. Alternatively, there is a matrix such that(7)is a diagonal quadratic form. The th column of the matrix is the vector .A nondegenerate symmetric bilinear form can be diagonalized, using Gram-Schmidt orthonormalization to find..

Composition theorem

Given a quadratic form(1)then(2)since(3)(4)

Sylvester's inertia law

The numbers of eigenvalues that are positive, negative, or 0 do not change under a congruence transformation. Gradshteyn and Ryzhik (2000) state it as follows: when a quadratic form in variables is reduced by a nonsingular linear transformation to the formthe number of positive squares appearing in the reduction is an invariant of the quadratic form and does not depend on the method of reduction.

Metric signature

The term metric signature refers to the signature of a metric tensor on a smooth manifold , a tool which quantifies the numbers of positive, zero, and negative infinitesimal distances of tangent vectors in the tangent bundle of and which is most easily defined in terms of the signatures of a number of related structures.Most commonly, one identifies the signature of a metric tensor with the signature of the quadratic form induced by on any of the tangent spaces for points . Indeed, given an orthogonal vector basis for any tangent space , the action of on arbitrary vectors and in is given by(1)whereby the signature of is defined to be the signature of any of the forms , i.e., the ordered triple of positive, negatives, and zero values for the inner products . This value is well-defined due to the fact that the signature of remains the same for all points in . For non-degenerate quadratic forms, the value will always satisfy , whereby the signature of will be..

Binary quadratic form discriminant

The discriminant of a binary quadratic formis defined byIt is equal to four times the corresponding binaryquadratic form determinant.Unfortunately, some authors define the discriminant of a binary quadratic form as the negative of above, so care is needed when consulting the literature.

Reduced binary quadratic form

The binary quadratic form is said to be reduced if the following conditions hold. Let be the discriminant, then 1. If is negative, is reduced if and if whenever or , and is called real. 2. If is positive, is reduced if , and is called imaginary or positive definite. Every imaginary binary quadratic form is equivalent to a unique reduced form and every real binary quadratic form is equivalent to a finite number of reduced forms.

Lorentzian space

Lorentzian -space is the inner product space consisting of the vector space together with the -dimensional Lorentzian inner product.In the event that the metric signature is used, Lorentzian -space is denoted ; the notation is used analogously with the metric signature .The Lorentzian inner product induces a norm on Lorentzian space, whereby the squared norm of a vector has the form(1)Rewriting (where by definition), the norm in (0) can be written as(2)In particular, the norm induced by the Lorentzian inner product fails to be positive definite, whereby it makes sense to classify vectors in -dimensional Lorentzian space into types based on the sign of their squared norm, e.g., as spacelike, timelike, and lightlike. The collection of all lightlike vectors in Lorentzian -space is known as the light cone, which is further separated into lightlike vectors which are positive and negative lightlike. A similar distinction is made for positive..

Binary quadratic form determinant

The determinant of a binary quadratic formis defined asIt is equal to 1/4 of the corresponding binaryquadratic form discriminant.Unfortunately, some authors define the determinant of a binary quadratic form as the negative of above, so care is needed when consulting the literature.

Quadratic form signature

The signature of a non-degenerate quadratic formof rank is most often defined to be the ordered pair of the numbers of positive, respectively negative, squared terms in its reduced form. In the event that the quadratic form is allowed to be degenerate, one may writewhere the nonzero components square to zero. In this case, the signature of is most often denoted by one of the triples or .A number of other, less common definitions are sometimes attributed to a quadratic form as its signature. In particular, the signature of is sometimes defined to be the number of positive squared terms in its reduced form, as well as the quantity .

Binary quadratic form

A binary quadratic form is a quadratic form intwo variables having the form(1)commonly denoted .Consider a binary quadratic form with real coefficients , , and , determinant(2)and . Then is positive definite. An important result states that there exist two integers and not both 0 such that(3)for all values of , , and satisfying the above constraint (Hilbert and Cohn-Vossen 1999, p. 39).

Quadratic form rank

For a quadratic form in the canonical formthe rank is the total number of square terms (both positive and negative).

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