Linear and multilinear algebra; matrix theory Matrix equations and groups Object

Unitary group

Description

Let $n \ge 1$ . The unitary group is the set $U(n) = \{U \in M_n(\C) : U^* U = I_n\}$ of complex $n \times n$ unitary matrices, with matrix multiplication. It is a subgroup of the general linear group $GL_n(\C)$ .

Remarks

It is the group of linear isometries of $\C^n$ with the standard inner product. $U(1)$ is the circle group $\{z \in \C : \abs{z} = 1\}$ . The unitary matrices with determinant $1$ form the special unitary group $SU(n)$ . $U(n)$ is a compact connected Lie group of real dimension $n^2$ .

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Identity matrix object
Matrix definition
Transpose definition
Conjugate transpose definition
Unitary matrix definition
Unitary group object

Tag	`0R9R0Y1HDK`
URI	`m2://0R9R0Y1HDK`
Kind	object
Writing	W1 stated
MSC	15B57, 20G20
Instance of	Group
Uses	Unitary matrix
Same as	Wikidata Q1500617 Wikipedia Unitary group

Unitary group

Description

Remarks

Reading path

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