Normal matrices with constraints on eigenvalues

Question

I need to find two different transformations with an inner product that follows these rules: (I don't study math in english so I'll try my best to explain)
1)
a. $TT^*=T^*T$ (normal)
b. All eigenvalues are in $\mathbb{R}$
c. $T\neq T^*$ (not self-adjoint)

2)
a. $TT^*=T^*T$ (normal)
b. All eigenvalues $\lambda$ satisfy $| \lambda| = 1 $
c. $TT^* \neq I$ ($T$ isn't unitary)

More than an answer I would be happy to hear an explanation as to how to approach something like this? (and not "brute forcing" every matrix I can think of and checking)

Answer 1

There aren't any such examples.

Every normal matrix $A$ can be written in the form $A=U\Lambda U^*$ where $U$ is unitary and $\Lambda$ is diagonal. The diagonal entries of $\Lambda$ are precisely the eigenvalues of $A$.

Therefore,

1. A normal matrix with real eigenvalues is self-adjoint
2. A normal matrix with unimodular eigenvalues is unitary.