# Unitary group

In mathematics, the unitary group of degree [itex]n[itex] over the field [itex]F[itex] (which is either the field [itex]\mathbb{R}[itex] of real numbers or the field [itex]\mathbb{C}[itex] of complex numbers) is the group of [itex]n[itex] by [itex]n[itex] unitary matrices with entries from [itex]F[itex], with the group operation that of matrix multiplication. This is a subgroup of the general linear group [itex]\mathrm{GL}(n,F)[itex].

In the simple case [itex]n=1[itex], the group [itex]\mathrm{U}(1)[itex] is the unit circle in the complex plane, under multiplication. All the complex unitary groups contain copies of this group.

If the field [itex]F[itex] is the field of real numbers then the unitary group coincides with the orthogonal group [itex]\mathrm{O}(n,\mathbb{R})[itex]. If [itex]F[itex] is the field of complex numbers one usually writes [itex]\mathrm{U}(n)[itex] for the unitary group of degree [itex]n[itex].

The unitary group [itex]\mathrm{U}(n)[itex] is a real Lie group of dimension [itex]n^2[itex]. The Lie algebra of [itex]\mathrm{U}(n)[itex] consists of complex [itex]n[itex]-by-[itex]n[itex] Skew-hermitian matrices, with the Lie bracket given by the commutator.

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy