Let the matrix $A \in \mathbb{R}^{n\times n}$ be diagonalizable $$ A = P^{-1} \Lambda P $$ Let $\widetilde{\lambda}$, $\widetilde{u}$ be an approximate eigenpair of $A$ with residual vector $r = A\widetilde{u} - \widetilde{\lambda}\widetilde{u}$, where $\|\widetilde{u}\|_2 = 1$. Then it is known that there exists an eigenvalue $\lambda$ such that $$ |\lambda - \widetilde{\lambda}| \le \text{cond}_2 P \|r\|_2, $$ where $\text{cond}_2 P$ is the condition number of $P$.
Also there is a notion of the condition number of a simple eigenvalue $$ \text{cond} (\lambda) = \dfrac{\|u\|_2 \|w\|_2}{|w^*u|}, $$ where $u$ and $w$ are the right and left eigenvectors, respectively, associated with $\lambda$. Moreover $$ \text{cond}_2(P) \ge \text{cond}(\lambda) $$ for any eigenvalue $\lambda$. It seems to me that the estimate $$ |\lambda - \widetilde{\lambda}| \le \text{cond}_2 P \|r\|_2 $$ is rather rough, since it provides the estimate using the whole matrix $P$, but not the individual condition number of the eigenvalue $\lambda$.
Are there any estimates of the form $$ |\lambda - \widetilde{\lambda}| \le \text{cond} (\lambda) \|r\|_2 ?$$
I will be glad to any books, articles, ideas and examples on this topic. Thank you in advance!