A Hermitian matrix $A$ has real eigenvalues. One way to check if it has a zero eigenvalue is to compute the determinant $\mathrm{det}(A)$. If $\mathrm{det}(A) = 0$, then $A$ has an eigenvalue of zero. However, computing the determinant, just as computing the eigenvalues, is a computationally expensive process.
A Hermitian matrix $A$ also has real diagonal entries. Is there any way to make use of this fact or other properties of a Hermitian matrix to quickly check if $A$ has an eigenvalue of zero?
