Are there any indictors to show whether or not our approximation of eigenvectors is good? There are researches that introduced algorithms for estimating the eigenvectors( If I am not mistaken Krylov subspace can be used to estimate eigenvectors of a matrix). Are there any indicators to show us whether or not our approximation is good?
Does any one know any books or articles that Discuss this topic?
 A: There are condition numbers for eigenvectors and more generally for invariant subspaces, which characterize sensitivity of eigenvectors with respect to small perturbations. 
Let $x$ is true eigenvector of th matrix $A$ associated with a simple eigenvalue $\lambda$ and let $\bar{x}$ be a true eigenvalue of the matrix $A + E$, where $E$ is a small perturbation. Let $angle(x,y)$ denote the angle between vectors $x$ and $y$. Then
$$angle(x,\bar{x}) \leq c(x)\|E\|_2 + o(\|E\|_2) \tag{1}$$
where $c(x)$ is the absolute condition number for eigenvector $x$ given by
$$c(x) = \|(T_{22} - \lambda I)^{-1}\|_2 = \sigma^{-1}_{min}(T_{22} - \lambda I) \tag{2}$$
assuming that $A = UTU'$ is the Schur factorization of the matrix $A$ ordered such that $T = \left[\begin{array}{cc}\lambda & T_{12} \\ 0 & T_{22}\end{array}\right]$.
In order to compute condition number (2) one needs whole Schur factorization $UTU'$ of the matrix $A$. Then one can construct error bound for the eigenvector associated with the first eigenvalue using (1) taking $E = A - UTU'$. The norm $\|E\|_2$ must be small enough, however there exist tests to check if (1) is valid for given $\|E\|_2$.
If only approximation of the eigenpair ($\hat{\lambda}$, $\hat{x}$) is available and the condition number (2) is known, then one can restrict $\|E\|_2$ using absolute backward error estimation, i.e.
$\|E\|_2 \leq \|A\hat{x} - \hat{\lambda}\hat{x}\|_2$ (it is assumed that $\|\hat{x}\|_2 = 1$). If approximation of eigenpair is obtained using the Krylov space methods, then one must use additional approximations for (2). 
Sensitivity analysis of eigenvectors is discussed in many books for example:


*

*Golub, Gene H., and Charles F. Van Loan. Matrix computations. Vol. 3. JHU Press, 2012.

*Saad, Youcef. Numerical methods for large eigenvalue problems. Vol. 158. Manchester: Manchester University Press, 1992.

*Kreßner, Daniel. "Numerical methods and software for general and structured eigenvalue problems." (2004).

