Quadratic form: Difference between revisions

Revision as of 09:35, 18 April 2023

Revision as of 01:51, 22 April 2023

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== 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]].

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

The case when all ''λ''_''i'' 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|indefinite]] (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 {{nowrap|(''p'', ''q'')}} (denoting ''p'' 1s and ''q'' −1s) is often denoted as '''R'''^''p'',''q'' 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 ''K''/(''K''^×)² (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)^{n_{-}}.</math>

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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''₀ 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 (resp., negative definite) if {{nowrap|''q''(''v'') > 0}} (resp., {{nowrap|''q''(''v'') < 0}}) for every nonzero vector ''v''.<ref>If a non-strict inequality (with ≥ or ≤) holds then the quadratic form ''q'' is called semidefinite.</ref> When ''q''(''v'') assumes both positive and negative values, ''q'' is an '''indefinite'''[[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 space|compact]]'' orthogonal group O(''n''). This stands in contrast with the case of indefiniteisotropic 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 ({{nowrap|1=O(''p'', ''q'') ≈ O(''q'', ''p''))}}, but the associated [[Clifford algebra]]s (and hence [[pin group]]s) are different.

== Definitions ==