TITLE: Number of independent components of a unitary matrix
QUESTION [10 upvotes]: By definition, a $n$ dimensional unitary matrix $U$ satisfies the condition
$U^{\dagger}U=I$,
and
$UU^{\dagger}=I$.
I'd like to ask if these two equations are independent. If so, there will be $n^2$ independent equations of constrain, which is equal to the number of independent components of a general $n$ dimensional matrix. This is obviously impossible. If not, how to prove it?

REPLY [2 votes]: Lemma.
A unitary $n \times n$ matrix has $n^2$ real independent parameters.
Proof. An arbitrary complex $n\times n$ matrix $U$ has $2n^2$ real independent parameters.
The unitarity condition $U^\dagger = U^{-1}$ is equivalent with both $UU^\dagger = I$, and $U^\dagger U = I$ (using the definition of an inverse matrix).
So any one of these conditions contains all constraints on $U$. We pick the last one.
\begin{equation}
U^\dagger U = I
\end{equation}
Taking the complex conjugate of the product on the left hand side shows that $(U^\dagger U)$ is an Hermitian matrix.
\begin{equation}
(U^\dagger U)^\dagger = U^\dagger (U^\dagger)^\dagger = U^\dagger U
\end{equation}
This may seem obvious since $I$ is clearly Hermitian, but that is not the point.
Knowing that the combination $U^\dagger U$ is Hermitian,
we want to know how much $U^\dagger U$ has to be constrained to not be just any Hermitian matrix, but the particular Hermitian matrix $I$.
Before such constraints are imposed,
the product $U^\dagger U$ (with same matrix twice) has the same $2n^2$ parameter-space as $U$.
Every subsequent constraint imposed on the parameters of $U^\dagger U$ must then correspond to a constraint on the parameters of $U$.
The total number of independent parameter constraints to make an arbitrary Hermitian matrix $U^\dagger U$ into $I$
is simply all of its parameters.
From the condition for an $n \times n$ Hermitian matrix (i.e. that $H^\dagger = H$) we find the number of free parameters. 
On the diagonal we require that the entries are purely real,
which sums to $n$ real parameters.
The conjugate symmetry condition on $H$ fixes all lower triangle elements in terms of the upper triangle (or vise versa).
The number of elements in the upper triangle is $n(n-1)/2$,
and each entry has $2$ real parameters.
Summing these terms gives us all independent parameters in a Hermitian matrix.
\begin{equation}
n + 2 \frac{n(n-1)}{2} = n^2
\end{equation}
So we remove $n^2$ parameters from $2n^2$
and we are done.
$\blacksquare$
References.
Credit goes to Yan Gobeil whose proof I adapted to this.