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Real quadratic forms

A fundamental problem is the classification of real quadratic forms under a linear change of variables.

Jacobi proved that, for every real quadratic form, there is an orthogonal diagonalization; that is, an orthogonal change of variables that puts the quadratic form in a " diagonal form "

where the associated symmetric matrix is diagonal. Moreover, the coefficients λ, λ,..., λ are determined uniquely up to a permutation.

If the change of variables is given by an invertible matrix that is not necessarily orthogonal, one can suppose that all coefficients λ are 0, 1, or −1. Sylvester's law of inertia states that the numbers of each 0, 1, and −1 are invariants of the quadratic form, in the sense that any other diagonalization will contain the same number of each. The signature of the quadratic form is the triple (n, n, n), where these components count the number of 0s, number of 1s, and the number of −1s, respectively. Sylvester's law of inertia shows that this is a well-defined quantity attached to the quadratic form.

The case when all λ have the same sign is especially important: in this case the quadratic form is called positive definite (all 1) or negative definite (all −1). If none of the terms are 0, then the form is called nondegenerate; this includes positive definite, negative definite, and isotropic quadratic form (a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a nondegenerate bilinear form. A real vector space with an indefinite nondegenerate quadratic form of index (p, q) (denoting p 1s and q −1s) is often denoted as R particularly in the physical theory of spacetime.

The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in K / (K) (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only "positive, zero, or negative". Zero corresponds to degenerate, while for a non-degenerate form it is the parity of the number of negative coefficients, (−1).

These results are reformulated in a different way below.

Let q be a quadratic form defined on an n -dimensional real vector space. Let A be the matrix of the quadratic form q in a given basis. This means that A is a symmetric n × n matrix such that

where x is the column vector of coordinates of v in the chosen basis. Under a change of basis, the column x is multiplied on the left by an n × n invertible matrix S, and the symmetric square matrix A is transformed into another symmetric square matrix B of the same size according to the formula

Any symmetric matrix A can be transformed into a diagonal matrix

by a suitable choice of an orthogonal matrix S, and the diagonal entries of B are uniquely determined – this is Jacobi's theorem. If S is allowed to be any invertible matrix then B can be made to have only 0, 1, and −1 on the diagonal, and the number of the entries of each type (n for 0, n for 1, and n for −1) depends only on A. This is one of the formulations of Sylvester's law of inertia and the numbers n and n are called the positive and negative indices of inertia. Although their definition involved a choice of basis and consideration of the corresponding real symmetric matrix A, Sylvester's law of inertia means that they are invariants of the quadratic form q.

The quadratic form q is positive definite if q (v) > 0 (similarly, negative definite if q (v) < 0) for every nonzero vector v. When q (v) assumes both positive and negative values, q is an isotropic quadratic form. The theorems of Jacobi and Sylvester show that any positive definite quadratic form in n variables can be brought to the sum of n squares by a suitable invertible linear transformation: geometrically, there is only one positive definite real quadratic form of every dimension. Its isometry group is a compact orthogonal group O(n). This stands in contrast with the case of isotropic forms, when the corresponding group, the indefinite orthogonal group O(p, q), is non-compact. Further, the isometry groups of Q and − Q are the same (O(p, q) ≈ O(q, p)), but the associated Clifford algebras (and hence pin groups) are different.