Can a Unitary matrix be non-diagonalizable?

Question

It is known that all non-diagonalizable matrices are non-hermitian. However, my question is that, are there some unitary matrices that are non-diagonalizable. What are the conditions ?

Answer 1

EDIT: the terminology doesn't seems standardized. Seeing wikipedia it says that a unitary matrix is an isometry on a finite dimensional complex vector space, however if you see in wikipedia unitary operator one of it examples is a rotation operator in $\mathbb{R}^2$.

So, if we assume that the underlying field is the complex field and the dimension of the vector space is finite we can pick the first definition stated above and so all unitary matrices will be diagonalizable.

However, if we go with the more general definition of unitary operator then we can say that not all unitary matrices are diagonalizable if the field is $\mathbb{R}$, by example many rotation matrices in $\mathbb{R}^2$ are unitary but doesn't have (real) eigenvalues, so they cannot be diagonalizable.