Properties of zero-diagonal symmetric matrices 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??

 A: Regarding your first two questions, the matrices that can be orthogonally transformed into a zero-diagonal symmetric matrix are exactly those symmetric matrices such that the sum of their eigenvalues is zero.
Indeed, since the trace of a symmetric matrix is the sum of its eigenvalues, the necessity follows. And the sufficiency follows from the Schur-Horn Theorem, that says that the possible diagonals of an operator are exactly those majorized by the eigenvalue vector; if the eigenvalues $\lambda_1,\ldots,\lambda_n$ add to zero, then the zero vector is majorized by $\lambda$ and so there is an orthonormal basis such that in that basis the operator has zero diagonal.
As for further properties of these matrices, I don't think much can be said: take any $n\times n$ symmetric matrix $A$ and expand it as $A\oplus\text{Tr}(-A)$; this is orthogonally similar to a zero diagonal matrix.
A: I'll consider the special case of symmetric tridiagonal matrices with zero diagonal for this answer.
I prefer calling the even-order tridiagonal ones Golub-Kahan matrices. These matrices turn up in deriving the modification of the QR algorithm for computing the singular value decomposition (SVD). More precisely, given an $n\times n$ bidiagonal matrix like ($n=4$)
$$\mathbf B=\begin{pmatrix}d_1&e_1&&\\&d_2&e_2&\\&&d_3&e_3\\&&&d_4\end{pmatrix}$$
the $2n\times 2n$ block matrix $\mathbf K=\left(\begin{array}{c|c}\mathbf 0&\mathbf B^\top \\\hline \mathbf B&\mathbf 0\end{array}\right)$ is similar to the Golub-Kahan tridiagonal
$$\mathbf P\mathbf K\mathbf P^\top=\begin{pmatrix}& d_1 &  &  &  &  &  &  \\d_1 &  & e_1 &  &  &  &  &  \\& e_1 &  & d_2 &  &  &  &  \\&  & d_2 &  & e_2 &  &  &  \\&  &  & e_2 &  & d_3 &  &  \\&  &  &  & d_3 &  & e_3 &  \\&  &  &  &  & e_3 &  & d_4 \\&  &  &  &  &  & d_4 & \end{pmatrix}$$
where $\mathbf P$ is a permutation matrix. This similarity transformation is referred to as the "perfect shuffle".
The importance of this is that the eigenvalues of the Golub-Kahan matrices always come in $\pm$ pairs; more precisely, if $\mathbf B$ has the singular values $\sigma_1,\sigma_2,\dots,\sigma_n$, then the eigenvalues of the Golub-Kahan tridiagonal are $\pm\sigma_1,\pm\sigma_2,\dots,\pm\sigma_n$.
Odd-order zero-diagonal tridiagonals can be treated similarly, as they have a zero eigenvalue in addition to the $\pm$ pairs of eigenvalues. The treatment given above for Golub-Kahan tridiagonals becomes applicable after deflating out the zero eigenvalue; one can do this by applying the QR decomposition $\mathbf T=\mathbf Q\mathbf R$ and forming the product $\mathbf R\mathbf Q$ and deleting the last row and last column, thus forming a Golub-Kahan tridiagonal.
See Ward and Gray's paper (along with the associated FORTRAN code) and this beautiful survey by David Watkins.
A: From an application standpoint:
If the rows and columns in the matrix M represent data objects, and the matrix entries are distances between them, then a Hermitian matrix H represents an asymmetric distance according to:
H = (1/2) (M+M^T) + i(1/2)(M-M^T)
where ^T is a matrix transpose
violating the axiom of a metric space that the distances d(A,B) = d(B,A)
for some "distance function" d()
The zero diagonal means that the distance from an object to itself is zero, preserving the axiom:
 d(A,A) = 0
