Given the eigendecomposition of a positive semidefinite, singular matrix $A$, is there a good way to determine if a small perturbation $A + \delta A$ is indefinite? Here, the perturbation $\delta A$ is also symmetric.
If $A$ and $\delta A$ shared the same eigenspace, then we could simply look at the eigenvalues and if the corresponding of eigenvalue of $\delta A$ to a zero eigenvalue of $A$ was negative, then the resulting perturbation would be indefinite. However, there are perturbations orthogonal to the eigenspace of $A$ that also cause the resulting sum to become indefinite and I'm not sure how to check those.
For example, consider the matrix $$ \begin{bmatrix} 1/2 & 1/2 & 0\\\\1/2 & 1/2 & 0\\\\0 & 0 & 1 \end{bmatrix} $$ with eigenvalues 0, 1, and 1. The perturbation $$ \begin{bmatrix} 1/\sqrt{2} & 0 & 0\\\\0 & -1/\sqrt{2} & 0\\\\0 & 0 & 0 \end{bmatrix} $$ causes $A + \epsilon \delta A$ to be indefinite for $\epsilon > 0$ and $\delta A$ is orthogonal to eigenspace of $A$. I'd like a way to check these cases.