Topological Space With Matrix Elements

Question

Let $U$ denote the set of all $n\times n$ matrices $A$ with complex entries such that $A$ is unitary(i.e $A^* A=I_n$). Then $U$ is a topological subspace of $C^{n^2}$,then which of the following is true?

1.Compact,but not connected

2.Connected,but not compact

3.Connected and compact

4.Neither connected nor compact

Answer 1

It is both compact and connected.

1. Compact, since every column of such matrix is unitary, and hence the matrix is bounded. In fact, $\|A\|_2=1$. The set is closed, as if $A_n$ are unitary and $A_n\to A$, then $A$ is also unitary.

2. Path Connectedness. If $A$ is a unitary matrix, then it can be written as $$ A=V^*\mathrm{diag}(\mathrm{e}^{ia_1},\ldots,\mathrm{e}^{ia_n})V $$ Then, the following is a continuous curve in $U$: $$ \varGamma(t)=V^*\mathrm{diag}(\mathrm{e}^{ia_1t},\ldots,\mathrm{e}^{ia_nt})V\in U $$ and $\varGamma(1)=A$, while $\varGamma(0)=I$.