Unitary Group

Overview

$U(n)$ and $SU(n)$ are the generalization of $O(n)$ and $SO(n)$ in ${\mathbb{C}}^n$

The inner product defined on ${\mathbb{C}}^n$ is as follows: For $u,v\in {\mathbb{C}}^n$:

$$u\cdot v=(u_1,u_2,\dots ,u_n)\cdot (v_1,v_2,\dots ,v_n)=u_1{\bar{v}}_1+u_2{\bar{v}}_2+\cdots +u_n{\bar{v}}_n$$

This is because we want $|v{|}^2$ to be a real number: $v\cdot v=v_1{\bar{v}}_1+v_2{\bar{v}}_2+\cdots +v_n{\bar{v}}_n=|v_1{|}^2+|v_2{|}^2+\cdots +|v_n{|}^2=|v{|}^2$ (see calculation rules - complex numbers)

Groups preserving this are called the unitary group $U(n)$ and the subgroup of $U(n)$ whose determinant is $1$ is the special unitary group $SU(n)$

Definition

$$U(n)=\{A\in M_n(\mathbb{C}):A^{\ast }A=I\}$$

Criterion for preserving the inner product

Similar to Rotation criterion:

Title

A linear transformation of ${\mathbb{C}}^n$ preserves the inner product if and only if its matrix $A$ satisfies $A{A}^{\ast }=I$ (see calculation rules - complex numbers)

Subgroup

The subgroup of $U(n)$ whose members have determinant $1$ is the special unitary group $SU(n)$

Example

• $SU(2)$ is the multiplication group of unit quaternions, or as matrix:

$$\left(\begin{array}{cc}\alpha & -\beta \\ \bar{\beta }& \bar{\alpha }\end{array}\right),\text{where }\alpha ,\beta \in \mathbb{C},|\alpha {|}^2+|\beta {|}^2=1$$ $$SU(2)\cong {\mathbb{S}}^3$$

• $U(1)$ or $U$ is the multiplication group of unit complex numbers
• $SU(1)$ is $\{1\}$

Path-Connectedness

$SU(n)$ is path-connected. proof similar to Path-Connectedness