Such a matrix: Except the main diagonal, the rest are Hermitian

Question

In a matrix, all the entries are complex numbers. If we set the main diagonal entries to be zero, then the matrix will be Hermitian.

Does this matrix has a name or some nice properties, especially about its eigenvalues???

Answer 1

I don't think there is a name for that matrix, and I think it's only property would be it is equal to its complex conjugate except for its diagonal. It's not going to be normal (i.e. it doesn't commute with it's complex conjugate), so it's not diagonalizable.

Likewise there's probably not much you'll be able to say much about the eigenvalues because the main result about the eigenvalues of a hermitian matrix is that all the eigenvalues must be real. This is not going to happen when you have complex entries on the diagonal.

It might be worth noting that changing the diagonal to any real numbers would also give you a hermitian matrix because the question tells us that all non-diagonal elements are symmetric with their complex conjugate and any real number is its own complex conjugate.

However, here are some properties of a zero-diagonal hermitian matrix you will end up with Properties of zero-diagonal symmetric matrices