I am trying to understand if bounding the operator norm (or L2 induced norm) of the difference of two matrices can be used to bound the eigenvalues (specifically, the min eigenvalue) of matrix products.
Formally, given full-rank matrices $Q \in \mathbb{R}^{n \times d}$ and $\hat{Q} \in \mathbb{R}^{n \times d}$. If eigenvalues of $Q^TQ$ are known (say $\lambda_1, \cdots, \lambda_d$) and that $||Q - \hat{Q}||_{2\rightarrow 2} = \epsilon$ (L2 induced norm or operator norm.) Can we say anything about the eigenvalues of the matrix $\hat{Q}^TQ$ in relation to $\lambda_1, \cdots, \lambda_n$?
My use case is that I know $\min_{1\leq i\leq d} \lambda_i > 0$ since $Q^TQ$ is PD. I am wondering if I can say anything about whether the minimum eigenvalue of $\hat{Q}^TQ$ will be negative. I understand that $\hat{Q}^TQ$ need not be symmetric and hence, the eigenvalues could be complex. From numerical experiments (using a symmetric $Q$ and $\hat{Q} = Q + \epsilon I$ with varying $\epsilon$ ensuring eigenvalues are always real) , I see that for sufficiently large $\epsilon$, the minimum eigenvalue becomes negative but I am curious if we can establish a relationship between $\epsilon$ and the minimum eigenvalue of $\hat{Q}^TQ$.
One way would be to define the perturbation $\epsilon V = (\hat Q - Q)^T Q$, so that you are interested in finding eigenvalues of $Q^T Q + \epsilon V$. For small $\epsilon$ you can approximate eigenvalues and eigenvectors using perturbation theory, it is a standard tool in quantum mechanics (https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics), https://en.wikipedia.org/wiki/Eigenvalue_perturbation#:~:text=In%20mathematics%2C%20an%20eigenvalue%20perturbation,to%20changes%20in%20the%20system. )