I want to express Q in terms of two complex variables $a$ and $b$, where Q is a $2\times 2$ orthogonal matrix.
So far I have got the following:
Let $Q = \left[\begin{smallmatrix} q_1 & q_2 \\ q_3 & q_4 \end{smallmatrix}\right]$. Using $Q^T Q = I$ gives:
$$\begin{bmatrix} q_1 & q_3 \\ q_2 & q_4 \end{bmatrix}\begin{bmatrix} q_1 & q_2 \\ q_3 & q_4 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$
$$\begin{bmatrix} q_1^2 + q_3^2 & q_1q_2 + q_3q_4 \\ q_1q_2 + q_3q_4 & q_2^2 + q_4^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$
When I try and solve this, I get $q_2^2 = q_3^2$ and $q_1^2 = q_4^2$. However, I am not sure where to go from here. I haven't used the fact that they are complex numbers so far, so I might need to use that?
I've realised from the comments that this wasn't worded very clearly-- I am working on some practise problems, and this is how the question is phrased:
"Assume that the orthogonal matrix A is $2\times 2$ with complex elements. Express the matrix elements of A in terms of two complex variables $a$ and $b$."
Hope that helps with clarity!