Complex Matrix is Orthogonal if and only if... Let D be a 2x2 matrix with entries in the complex numbers. Prove that D is orthogonal if and only if, it is of the form:
\begin{pmatrix}
a & -b\\ 
b & a
\end{pmatrix}
or
\begin{pmatrix}
a & b\\ 
b & -a
\end{pmatrix}
Proof. I've already proved that if D is equal to those forms then, it implies that D is an orthogonal matrix. But how can I prove this? If D is orthogonal, then it must be of the form:
\begin{pmatrix}
a & -b\\ 
b & a
\end{pmatrix}
or
\begin{pmatrix}
a & b\\ 
b & -a
\end{pmatrix}
Update: a and b must satisfy that $a^2+b^2=1$

If $A= \begin{pmatrix}a & b \\ 
c & d\end{pmatrix}$
then we get from $A^TA= I$ that $a^2+c^2=1= b^2+d^2$ and also $ab+cd = 0$. Then $ab = -cd$ so squaring it gives $a^2b^2 = c^2d^2$. I'll let you finish the rest.
