Editing Quadratic form (section)

== Real quadratic forms ==
{{see also|Sylvester's law of inertia|Definite quadratic form|Isotropic quadratic form}}
A fundamental problem is the classification of real quadratic forms under a [[linear transformation|linear change of variables]].

[[Carl Gustav Jacobi|Jacobi]] proved that, for every real quadratic form, there is an [[orthogonal diagonalization]]; that is, an [[orthogonal transformation|orthogonal change of variables]] that puts the quadratic form in a "[[diagonal form]]"
<math display="block"> \lambda_1 \tilde x_1^2 + \lambda_2 \tilde x_2^2 + \cdots + \lambda_n \tilde x_n^2, </math>
where the associated symmetric matrix is [[diagonal matrix|diagonal]]. Moreover, the coefficients {{math|''λ''<sub>1</sub>, ''λ''<sub>2</sub>, ..., ''λ''<sub>''n''</sub>}} are determined uniquely [[up to]] a [[permutation]].<ref>[[Maxime Bôcher]] (with E.P.R. DuVal)(1907) ''Introduction to Higher Algebra'', [https://babel.hathitrust.org/cgi/pt?id=uc1.b4248862;view=1up;seq=147 § 45 Reduction of a quadratic form to a sum of squares] via [[HathiTrust]]</ref>

If the change of variables is given by an [[invertible matrix]] that is not necessarily orthogonal, one can suppose that all coefficients {{math|''λ''<sub>''i''</sub>}} are 0, 1, or −1. [[Sylvester's law of inertia]] states that the numbers of each 0, 1, and −1 are [[invariant (mathematics)|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 {{math|(''n''<sub>0</sub>, ''n''<sub>+</sub>, ''n''<sub>−</sub>)}}, where these components count the number of 0s, number of 1s, and the number of −1s, respectively. [[James Joseph Sylvester|Sylvester]]'s law of inertia shows that this is a well-defined quantity attached to the quadratic form.

The case when all {{math|''λ''<sub>''i''</sub>}} have the same sign is especially important: in this case the quadratic form is called '''[[positive definite form|positive definite]]''' (all 1) or '''negative definite''' (all −1).  If none of the terms are 0, then the form is called '''{{visible anchor|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 form|nondegenerate ''bilinear'' form]]. A real vector space with an indefinite nondegenerate quadratic form of index {{math|(''p'', ''q'')}} (denoting {{math|''p''}} 1s and {{math|''q''}} −1s) is often denoted as {{math|'''R'''<sup>''p'',''q''</sup>}} particularly in the physical theory of [[spacetime]].

The [[Discriminant#Discriminant of a quadratic form|discriminant of a quadratic form]], concretely the class of the determinant of a representing matrix in {{math|''K'' / (''K''<sup>×</sup>)<sup>2</sup>}} (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, {{math|(−1)<sup>''n''<sub>−</sub></sup>}}.

These results are reformulated in a different way below.

Let {{math|''q''}} be a quadratic form defined on an {{math|''n''}}-dimensional [[real number|real]] vector space. Let {{math|''A''}} be the matrix of the quadratic form {{math|''q''}} in a given basis. This means that {{math|''A''}} is a symmetric {{math|''n'' × ''n''}} matrix such that
<math display="block">q(v) = x^\mathsf{T} Ax,</math>
where ''x'' is the column vector of coordinates of {{math|''v''}} in the chosen basis. Under a change of basis, the column {{math|''x''}} is multiplied on the left by an {{math|''n'' × ''n''}} [[invertible matrix]] {{math|''S''}}, and the symmetric square matrix {{math|''A''}} is transformed into another symmetric square matrix {{math|''B''}} of the same size according to the formula
<math display="block"> A\to B=S^\mathsf{T}AS.</math>

Any symmetric matrix {{math|''A''}} can be transformed into a diagonal matrix
<math display="block"> B=\begin{pmatrix}
\lambda_1 & 0 & \cdots & 0\\
0 & \lambda_2 & \cdots & 0\\
\vdots & \vdots & \ddots & 0\\
0 & 0 & \cdots & \lambda_n
\end{pmatrix}</math>
by a suitable choice of an orthogonal matrix {{math|''S''}}, and the diagonal entries of {{math|''B''}} are uniquely determined – this is Jacobi's theorem. If {{math|''S''}} is allowed to be any invertible matrix then {{math|''B''}} can be made to have only 0, 1, and −1 on the diagonal, and the number of the entries of each type ({{math|''n''<sub>0</sub>}} for 0, {{math|''n''<sub>+</sub>}} for 1, and {{math|''n''<sub>−</sub>}} for&nbsp;−1) depends only on&nbsp;{{math|''A''}}. This is one of the formulations of Sylvester's law of inertia and the numbers {{math|''n''<sub>+</sub>}} and {{math|''n''<sub>−</sub>}} 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&nbsp;{{math|''A''}}, Sylvester's law of inertia means that they are invariants of the quadratic form&nbsp;{{math|''q''}}.

The quadratic form {{math|''q''}} is positive definite if {{math|''q''(''v'') > 0}} (similarly, negative definite if {{math|''q''(''v'') < 0}}) for every nonzero vector {{math|''v''}}.<ref>If a non-strict inequality (with ≥ or ≤) holds then the quadratic form {{math|''q''}} is called semidefinite.</ref> When {{math|''q''(''v'')}} assumes both positive and negative values, {{math|''q''}} is an [[isotropic quadratic form]]. The theorems of Jacobi and [[James Joseph Sylvester|Sylvester]] show that any positive definite quadratic form in {{math|''n''}} variables can be brought to the sum of {{math|''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 space|compact]]'' orthogonal group {{math|O(''n'')}}. This stands in contrast with the case of isotropic forms, when the corresponding group, the [[indefinite orthogonal group]] {{math|O(''p'', ''q'')}}, is non-compact. Further, the isometry groups of {{math|''Q''}} and {{math|−''Q''}} are the same ({{math|1=O(''p'', ''q'') ≈ O(''q'', ''p''))}}, but the associated [[Clifford algebra]]s (and hence [[pin group]]s) are different.