Unitary group

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

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

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

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

See also: Special unitary group