I am trying to prove that every unitary $2 \times 2$ matrix has the form
\begin{bmatrix}
a & -\overline{b} \\
b & \overline{a}
\end{bmatrix}
where $a, b \in \mathbb{C}$ such that $|a|^{2} + |b|^{2} = 1$
Assuming that $U$ is a unitary matrix, then it follows $$U^{*}U = I = UU^{*} $$
$$\pmatrix{x & y\\w & z} \pmatrix{\overline{x} & \overline{w}\\\overline{y} & \overline{z}} = \pmatrix{1 & 0\\0 & 1}$$
which means
$$x\overline{x}+w\overline{w} = |x|^{2} + |w|^{2}= 1$$
From matrix multiplication above we also get the equations:
$$w\overline{x} + z\overline{y} = 0$$
$$x\overline{w} + y\overline{z} = 0 $$
I kept trying different substitutions, but I am still stuck at proving that $z = \overline{x}$ and $y = -\overline{w}$
Help?
Best Answer
You can't prove that because it is not true. All the matrices of the form$$\begin{bmatrix}a&-\overline b\\b&\overline a\end{bmatrix}$$such that $|a|^2+|b|^2=1$ have determinant $1$. The unitary $2\times2$ matrices are of the form$$\begin{bmatrix}a&-\omega\overline b\\b&\omega\overline a\end{bmatrix},$$with $|a|^2+|b|^2=1$ and $|\omega|=1$.