Does there exist a normal matrix which is not diagonal, nor Hermitian nor Unitary nor Skew Hermitian?

Question

Well, we know that if a matrix is diagonal, Hermitian, Unitary or Skew-hermitian, then the matrix is Normal.

But is the converse true? If not, can anyone give an example?

Answer 1

If you want a real normal matrix that is not a scalar multiple of a diagonal, Hermitian, skew-Hermitian or unitary matrix, it can be shown that the smallest-sized example is $3\times3$. As pointed out by a comment, there is a nice $3\times3$ example in Wikipedia.

For non-real matrices, one simple example is $\pmatrix{2+i&2-i\\ 2-i&2+i}$. Basically, if you take a random unitary matrix $U$ and a random complex diagonal matrix $D$, the product $UDU^\ast$ will almost certainly satisfy your need.