I'm interested in the properties of zero-diagonal symmetric (or hermitian) matrices, also known as hollow symmetric (or hermitian) matrices.
The only thing I can come up with is that it cannot be positive definite (if it's not the zero matrix): The Cholesky decomposition provides for positive definite matrices $A$ the existence of a lower triangular $L$ with $A=LL^*$. However the diagonal of $LL^*$ is the inner product of each of the rows of $L$ with itself. Since the diagonal of $A$ consists of zeros, so $L$ (and thus $A$) must be the zero matrix.
The sorts of questions that interest me are:
- which symmetric matrices can be transformed orthogonally into a zero-diagonal matrix?
- what can we say about the eigen-values of a zero-diagonal symmetric matrix?.
- any other interesting known properties??