How would the most general $2 \times 2$ normal matrix look like

Question

How would the most general $2 \times 2$ normal matrix look like?

The normal matrix satisfy equation: $A^*A=AA^*$ where $A^*$ denotes
conjugate transpose.

I was thinking about the matrix:

$$
\begin{pmatrix}
a & -b \\
b & a \\
\end{pmatrix}
$$ because its columns are orthogonal to each other and it satisfies the given equation:

$$
\begin{pmatrix}
a & -b \\
b & a \\
\end{pmatrix}
\begin{pmatrix}
a & b \\
-b & a \\
\end{pmatrix} =
\begin{pmatrix}
a^2 + b^2 & 0 \\
0 & b^2 + a^2 \\
\end{pmatrix}
$$
$$
$$

$$
\begin{pmatrix}
a & b \\
-b & a \\
\end{pmatrix}
\begin{pmatrix}
a & -b \\
b & a \\
\end{pmatrix} =
\begin{pmatrix}
a^2 + b^2 & 0 \\
0 & b^2 + a^2 \\
\end{pmatrix}
$$
$$
$$

It is true for real matrices, and I suppose for the complex one too. But is this the most general case, or is there something else?

Answer 1

You can also characterize a normal matrix $A$ by the fact there exists an unitary matrix such that $U^{-1}AU=D$ where $D$ is diagonal.

Unitary $2x2$ matrices are well characterized, see for instance the wiki page: https://en.wikipedia.org/wiki/Unitary_matrix

$\exists (\alpha,\beta,\xi,\zeta)\in\mathbb R^4$ such that $U=\begin{bmatrix}\cos \alpha & -\sin \alpha\\\sin \alpha & \cos \alpha\end{bmatrix}\begin{bmatrix}e^{i\xi} & 0\\0 & e^{i\zeta}\end{bmatrix}\begin{bmatrix}\cos \beta & -\sin \beta\\\sin \beta & \cos \beta\end{bmatrix}$

$U=R(\alpha).\operatorname{diag}(e^{i\xi},e^{i\zeta}).R(\beta)$

$U^{-1}=R(-\beta).\operatorname{diag}(e^{-i\xi},e^{-i\zeta}).R(-\beta)$

Now you can take $D=\begin{bmatrix}z_1 & 0\\0 & z_2\end{bmatrix}$

Multiplying all these won't probably be very nice looking, but you get to parametrize any normal matrix with $4$ reals and $2$ complexes.

Yet, somehow, this isn't very satisfying for me, the dimension on $\mathbb R$ still appear to be $8$ while C.Haattingh showed that we need $|b|^2=|c|^2$ which can be rewritten $b=re^{it_b}$ and $c=re^{it_c}$ and we reduced the dimension by $1$ already. So I suspect that my $D$ and $(\xi,\zeta)$ are in fact not independent... Is it possible to fix that ?