<s>[INST] How would the most general $2 \times 2$ normal matrix look like? 
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?
 [/INST] Let $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}.$$ For $A$ to be normal we require that $$\begin{bmatrix} \overline{a} & \overline{c} \\ \overline{b} & \overline{d} \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} \overline{a} & \overline{c} \\ \overline{b} & \overline{d} \end{bmatrix},$$ or equivalently $$\begin{bmatrix} |a|^2 + |c|^2 & \overline{a}b + \overline{c}d \\ \overline{b}a + \overline{d}c & |b|^2 + |d|^2 \end{bmatrix} = \begin{bmatrix} |a|^2 + |b|^2 & a\overline{c} + b\overline{d} \\ c\overline{a} + d\overline{b} & |c|^2 + |d|^2 \end{bmatrix}.$$
Firstly notice that by equality of the diagonal entries we must have $$|b|^2 = |c|^2.$$ And furthermore comparing the off-diagonal entries we must have 
$$ \overline{a}b + \overline{c}d = a\overline{c} + b\overline{d}, $$
$$ \overline{b}a + \overline{d}c = c\overline{a} + d\overline{b}. $$
In the real case the first requirement reduces to $c=|b|$, but for complex matrices we could also have for example $b=1$ and $c=i$. For now let's just consider the real case: the most general case is to fix $b$ and let $c=|b|$. Then we have two cases:


*

*If $b$ is nonnegative $a$ and $d$ could be any real values.

*If $b$ is negative $a=d$.  


The second case above corresponds to your example - I wouldn't consider this to be more general than the first case - for real numbers 1 and 2 together is most general.
So now for complex numbers there is a lot more work to be done: you could explore this further by for example setting $a = a_1 + ia_2, b=b_1 + ib_2, \ldots$, fixing some and then working out by the constraints what values the other variables could assume...
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