All Science Fair Projects

Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

Unitary group

(Redirected from U(1))

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

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

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

The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of complex n-by-n Skew-hermitian matrices, with the Lie bracket given by the commutator.