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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??
  • 2
    The eigenvalues sum up to zero.2011-11-07
  • 2
    Nice, thanks (: Since $tr(AB)=tr(BA)$ and if $U$ is an o-n basis of eigen vectors of $A$ then $tr(A)=tr(U^*AU)$ which is the sum of the eigen values of $A$.2011-11-07
  • 0
    If the matrix is not null matrix, it should have +ve as well as -ve eigenvalues i.e., it is indefinite matrix.2011-11-07

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