Let $A$, $B$ and $E$ be three symmetric $n \times n$ matrices where $B = A+E$. Let $\mathcal{Q}$ be the set of orthonormal matrices that diagonalize $A$, i.e. the set of orthogonal matrices $Q$ such that $Q^TAQ = D$ where $D$ is diagonal. Let $\mathcal{P}$ be the set of orthogonal matrices that diagonalize $B$, i.e. the set of orthogonal matrices $P$ such that $P^TBP = S$ where $S$ is diagonal.
Do there exists appropriate matrices $Q \in \mathcal{Q}$ and $P \in \mathcal{P}$ such that there an upper bound on the Frobenius norm $\|Q-P\|_F$ in terms of $\|B-A\|_F =\|E\|_F$ times some appropiate constant $C$?
Notice that this question asks a similar thing, but only considers the case when $E$ has very small norm, whereas I am interested in an upper bound given by the norm of $E$. In case of an affirmative answer, does there exist as well an upper bound for $\|Q-P\|_1$ in terms of $\|E\|_1 $ ? By norm 1 of $E$ we mean the sum of the absolute value of the entries of $E$.
If it helps, we may assume $A$ and $B$ have no repeated eigenvalues.
There is no constant $C$ that works for all such $A$ and $B$. Note that for any nonzero $t$, $Q$ diagonalizes $A$ iff it diagonalizes $tA$. Thus for any $C$, and $A$ and $B$ that are not simltaneously diagonalizable, you can take $t > 0$ sufficiently small that there are no orthogonal matrices $P$ and $Q$ that diagonalize $tA$ and $tB$ respectively with $\|P - Q\|_F \le C \|tA - tB\|_F = C t \|A-B\|_F$.