Determine all $2 \times 2$ normal matrices.
In particular, how would I show that there are normal matrices which are neither unitary, Hermitian, skew-Hermitian, symmetric, nor skew-symmetric. The only thing I do know is $AA^* = A^*A$, but I'm not sure how to proceed. I tried writing in $a,b,c,d$ as entries of $2 \times 2$ matrix, but it seemed to lead nowhere.
Best Answer
Note that $A$ is normal if and only if $AA^* - A^*A = 0$. If we take $$ A = \pmatrix{a&b\\c&d}, $$ then we find $$ AA^* - A^*A = \pmatrix{|b|^2 - |c|^2 & -b \bar a + a \bar c + b \bar d - d \bar c\\ \overline{-b \bar a + a \bar c + b \bar d - d \bar c} & |c|^2 - |b|^2}. $$ $A$ will be normal if and only if all the above entries are zero. In particular, we see that this means that $A$ is normal if and only if $$ |c| = |b|, \quad -b \bar a + a \bar c + b \bar d - d \bar c = 0. $$ In order to take advantage of the first equation, write $b,c$ in polar form. That is, let's say $$ b = r_1e^{i\theta}, \quad c = r_1e^{i \phi} $$ where $r_1 \geq 0$ and $\theta,\phi \in \Bbb R$. Substituting these into the second equation yields $$ - r_1e^{i \theta}\bar a + a r_1e^{-i\phi} + re^{i\theta} \bar d - d r_1e^{-i\phi} = 0 \implies\\ e^{i \theta}[\bar d-\bar a] + e^{-i \phi}[a-d] = 0 \implies\\ e^{i \theta}[\overline{d-a}] - e^{-i \phi}[d-a] = 0 \implies\\ e^{i (\theta + \phi)}[\overline{d-a}] - [d-a] = 0. $$ Write $d-a$ in polar form. That is, take $d-a = r_2 e^{i \psi}$. We can write the above as the equation $$ e^{i (\theta + \phi)}r_2 e^{-i\psi} - r_2e^{i\psi} = 0 \implies\\ e^{i(\theta + \phi - \psi)} = e^{i \psi} \implies\\ e^{i(\theta + \phi)} = [e^{i \psi}]^2 \implies\\ \pm \exp\left[i\frac{\theta + \phi}2\right] = e^{i \psi} = \frac{d-a}{r_2} \implies\\ d-a = \pm r_2 \exp\left[i\frac{\theta + \phi}2\right]. $$
The above analysis leads to the following parameterization of the normal matrices. Select any $a \in \Bbb C$, $r_1,r_2 \geq 0$ and angles $\phi,\psi$. The matrix with this value of $a$ and $$ b = r_1e^{i\theta}, \quad c = r_1e^{i \phi}, \quad d = a \pm r_2 \exp\left[i\frac{\theta + \phi}2\right] $$ is necessarily normal, and every normal matrix can be written in this way. We could also tweak some definition to get the equivalent parameterization $$ a,b = [\text{arbitrary complex}], \quad c = b e^{2i k_1}, \quad d = a + k_2 e^{i k_1} $$ where $k_1,k_2$ are real. This amounts to $$ A = aI + \pmatrix{0&b\\b e^{2i k_1} & k_2 e^{ik_1}}. $$
As far as the second part goes: it suffices to take a multiple of a unitary matrix. For instance, take the matrix of any rotation by an angle that is not a multiple of $\pi/2$, and multiply it by some $\alpha > 1$.
Alternatively: for any normal $A$, there exists a $\gamma \in \Bbb C$ such that $A + \gamma I$ is neither unitary, Hermitian, nor skew-Hermitian. To see that this is the case, it suffices to look at the expressions for $M^*M$ and $M \pm M^*$, where $M = A + \gamma$ (in terms of $A$ and $\gamma$, not in terms of the entries).
Alternatively: note that every diagonal matrix is normal. However, not every diagonal matrix is Hermitian, skew-Hermitian, or unitary.