Suppose I have some matrix $A$ and I perturb it by some small amount: $A \to A + \delta A$. I am interested in determining the sign of the quantity $a^\dagger \delta A a$. Here is what I know:
- $a$ is an eigenvector of the unperturbed matrix $A$. Specifically, it is the eigenvector corresponding to the most negative eigenvalue of $A$.
- The diagonal elements of $\delta A$ are all positive.
- $A$ is Hermitian, and so is $\delta A$
- $A$ is an operator (it is infinite-dimensional)
Is it possible to determine the sign of the quantity of interest? Unfortunately, I don't know anything about the signs of the off-diagonal components, so I do not believe I can make any arguments about the positive-definiteness of $\delta A$. Thank you very much for the help! I am a physicist, so apologies in advance if this question is too vague or poorly-worded.
EDIT: to get $\delta A$, you suppose that $A$ is initially parameterized by some variable $q$. Then, you take the variation: $\delta A = \frac{\partial A}{\partial q} \delta q$