📂Abstract Algebra

Unitary Group

Definition

$n \times n$ The set of unitary matrices is denoted as $\mathrm{U}(n)$ and is called the unitary group of degree $n$.

$$\mathrm{U}(n) := \left\{ n \times n \text{ unitary matrix} \right\} = {\left\{ A \in M_{n \times n}(\mathbb{C}) : A A^{\ast} = I \right\}}$$

Here, $A^{\ast}$ is the conjugate transpose matrix.

Explanation

Since it only collects unitary matrices, it forms a group under matrix multiplication. It is a subgroup of the general linear group $\mathrm{GL}(n, \mathbb{C})$.

It is a Lie group because it has a differentiable structure.