# The Unitary Group (Lie Groups)

For any complex matrix $A$, the adjoint of A is the matrix $A*$ formed by conjugating the entries of $A$ and taking the transpose:

$A* = \bar{A}^{T}$. Observe that $(AB)* = (\bar{A}\bar{B})^{T} = \bar{B}^{T}\bar{A}^{T} = B*A*$. For any positive integer $n$, the unitary group of degree n is the subgroup $U(n) \subseteq GL(n, \mathbb{C})$ consisting of complex $n \times n$ matrices whose columns form an orthonormal basis for $\mathbb{C}^{n}$ with respect to the Hermetian dot product.

We will show that $U(n)$ is a properly embedded Lie subgroup of $GL(n, \mathbb{C})$ of dimension $n^{2}$.

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