# Difference between the leading eigenvalues of two matrices

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.

• The difference is bound by $\epsilon$ as long as both matrices are normal, see the Bauer-Fike theorem. – Conifold Jan 2 at 4:00

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}