Suppose we have two real, symmetric and positive semidefinite matrices $A$ and $B$, and we know that they approximate each other well in the sense that
$$\| A – B \|_2 \le \epsilon,$$
where $\epsilon$ is positive and $\|\cdot\|_2$ is the $\ell_2$ operator norm. Is it possible to bound the difference between their leading eigenvalues, $|\lambda_1(A) – \lambda_1(B)|$, in terms of $\epsilon$?
I know it's possible to bound the angle between the leading eigenvectors of $A$ and $B$, but I have no idea how to prove a similar bound for eigenvalues.
Best Answer
We may assume that $\lambda_\max(A)\ge\lambda_\max(B)$. Then \begin{aligned} |\lambda_\max(A)-\lambda_\max(B)| &=\lambda_\max(A)-\lambda_\max(B)\\ &=\|A\|_2-\|B\|_2\\ &\le\|A-B\|_2 \le\epsilon. \end{aligned}