# Preconditioned GMRES for nearly diagonalizable systems

I have been working with a matrix $$A$$ and preconditioner $$P\approx A^{-1}$$ that I've then applied GMRES to the (left) preconditioned linear system

$$$$P^{-1}Ax=P^{-1}b$$$$

$$P^{-1}A$$ is not normal and so cannot be diagonalized by a unitary matrix. I have been able to show that for arbitrarily small perturbations $$\varepsilon$$, $$P(\varepsilon)^{-1}A(\varepsilon)$$ is diagonalizable. $$P(\varepsilon)$$, $$A(\varepsilon)$$ are both continuous functions of $$\varepsilon$$ at $$\varepsilon=0$$. For the purposes here, imagine $$A(\varepsilon)=A+\varepsilon I$$, $$P(\varepsilon)=P+\varepsilon I$$.

From Proposition 4 of Saad and Schultz's 1986 GMRES paper I can characterize the rate at which the linear system residuals decrease for the $$\varepsilon>0$$ perturbed system

$$$$P(\varepsilon)^{-1}A(\varepsilon)x=P(\varepsilon)^{-1}b,$$$$

for arbitrarily small but nonzero $$\varepsilon$$. Can this be used to inform how the linear system residuals decrease for the unperturbed system

$$$$P^{-1}Ax=P^{-1}b?$$$$

My initial intuition is that, while $$\varepsilon$$ is much smaller than the norm of the linear system residual that the perturbed system can be used to characterize the rate of convergence for the unperturbed system.

This can't be the case though since a nondiagonalizable matrix is always arbitrarily close to a matrix with distinct eigenvalues which is diagonalizable.

Any feedback would be greatly appreciated.