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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