I understand that the dimension of the $SU(n)$ matrix group is $n^2$ because there are $2n^2$ real variables (for the $n^2$ complex matrix elements) in each matrix, and there are $n^2$ equations (arising from the unitary condition) that relate the $2n^2$ real variables, so that the number of independent real parameters drops down to $n^2$. This is the dimension of the $U(n)$ group. The dimension of the $SU(n)$ group is $n^2 -1$ because an additional constraint of unit determinant reduces the number of independent parameters down by one.
My question is this:
Say we have a matrix $R = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Then, $R^\dagger = \begin{pmatrix} a^* & c^* \\ b^* & d^* \end{pmatrix}$. Therefore, $R^\dagger R=1$ implies that $|a|^2 + |b|^2 = 1$, $|c|^2 + |d|^2 = 1$, $a^*b + c^*d = 0$, and $ab^* + cd^* = 0$.
I was wondering if the last two equations are the same. If so, then the number of independent parameters does not really go down from $2n^2$ to $n^2$, does it?
They are equivalent complex equations, but they are also expressible as two separate real equations, when you examine the real part and the complex part. Thus, you still have $4 = n^2$ conditions to be satisfied.