Create a Unitary 2x2 matrix given a complex number "a".

Question

I am trying to create a 2x2 Unitary matrix ($U^\dagger U=1$) with a complex number "a". The absolute value of the the number can also be greater than 1. The matrix's first element has to be 'a'.

I thought that

\begin{pmatrix} a & \sqrt{1-|a|^2}\\ \sqrt{1-|a|^2} & -a^* \end{pmatrix}

would do the job, but it turns out that if |a|>1 it does not work.

Any help will be appreciated.

Answer 1

There is a general expression of any $2\times2$ unitary matrix. If $a$ is the first element, the matrix has to be of the form

$U = \begin{bmatrix} a & b \\ -z_0 \overline{b} & z_0 \overline{a} \end{bmatrix}$

with $|z_0|=1$ and $|a|^2+|b|^2=1$. This last equation (coming from the fact that $|det(U)|=1$) imposes that $|a| \le 1$.