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 $n^2$.

Explanations

L1 human

The unitary group collects all complex matrices that keep the length of every vector. For $n = 1$, these are multiplications by complex numbers of absolute value $1$, that is, rotations of the complex plane. The group appears throughout physics.