Suppose I have a symmetric positive semidefinite matrix $A$ with leading eigenvalue $1$ of multiplicity $1$ and remaining eigenvalues $\leq\epsilon$. I am told that another symmetric positive semidefinite matrix $B$ is close in some sense to $A$. I wish to conclude that the leading eigenvector of $B$ is therefore close in some sense to that of $A$.

**Question:** What notions of closeness correspond to existing theorems of this sort?

One solution: Let $a$ and $b$ denote leading eigenvectors of $d\times d$ matrices $A$ and $B$, respectively, scaled so that $\|a\|_2^2=1$ and $\|b\|_2^2$ is the leading eigenvalue of $B$. Then by triangle, Eckart–Young–Mirsky, and triangle again, we have

$$\|bb^\top-aa^\top\|_F\leq\|bb^\top-B\|_F+\|B-aa^\top\|_F\leq2\|B-aa^\top\|_F\leq2\big(\|B-A\|_F+\|A-aa^\top\|_F\big)\leq2\|B-A\|_F+2\sqrt{d-1}\epsilon.$$

For this attempt, I don't like the additive loss in $\epsilon$, and I doubt it's necessary. Can I get better performance with a different norm? Perhaps the spectral norm?