Editing Unitary group (section)

=== Indefinite forms ===
Analogous to the [[indefinite orthogonal group]]s, one can define an '''indefinite unitary group''', by considering the transforms that preserve a given Hermitian form, not necessarily positive definite (but generally taken to be non-degenerate). Here one is working with a [[vector space]] over the complex numbers.

Given a Hermitian form Ψ on a complex vector space ''V'', the unitary group U(Ψ) is the group of transforms that preserve the form: the transform ''M'' such that {{nowrap|1=Ψ(''Mv'', ''Mw'') = Ψ(''v'', ''w'')}} for all {{nowrap|''v'', ''w'' ∈ ''V''}}. In terms of matrices, representing the form by a matrix denoted Φ, this says that {{nowrap|1=''M''<sup>∗</sup>Φ''M'' = Φ}}.

Just as for [[Symmetric bilinear form|symmetric forms]] over the reals, Hermitian forms are determined by [[Signature of a quadratic form|signature]], and are all [[Matrix congruence|unitarily congruent]] to a diagonal form with ''p'' entries of 1 on the diagonal and ''q'' entries of −1. The non-degenerate assumption is equivalent to {{nowrap|1=''p'' + ''q'' = ''n''}}. In a standard basis, this is represented as a quadratic form as:
: <math>\lVert z \rVert_\Psi^2 = \lVert z_1 \rVert^2 + \dots + \lVert z_p \rVert^2 - \lVert z_{p+1} \rVert^2 - \dots - \lVert z_n \rVert^2</math>

and as a symmetric form as:
: <math>\Psi(w, z) = \bar w_1 z_1 + \cdots + \bar w_p z_p - \bar w_{p+1}z_{p+1} - \cdots - \bar w_n z_n.</math>

The resulting group is denoted {{nowrap|U(''p'',''q'')}}.