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??

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.

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.