Error Bound for Computed Eigenvalues.

From (7.105) and the perturbation theory of eigenvalues and eigenvectors, it can shown that up to the first order of residual norms $\Vert r\Vert _2$ and $\Vert s^{\ast}\Vert _2\/$, there is an eigenvalue $\lambda$ of $A$ satisfying

$$\vert\lambda - \wtd\lambda \vert \simle \frac{1}{\vert\wtd ... ... \max\left\{\Vert r\Vert _2, \Vert s^{\ast} \Vert _2 \right\},$$ (212)

where ``$\simle$'' denotes ``less than'' up to the first order of $\Vert r\Vert _2$ and $\Vert s^{\ast}\Vert _2\/$. Note that the individual condition number $c_{\lambda}$ is defined as

$$c_{\lambda}\equiv \frac {1}{\vert y^{\ast} x\vert} =\sec\theta(x,y),$$

and $x$ and $y$ are the corresponding eigenvector and left eigenvector of $A$ and are normalized so that $\Vert x\Vert _2 = \Vert y\Vert _2 = 1$. Since exact $x$ and $y$ usually are not known, for all practical purposes, one could simply estimate this condition number $c_{\lambda}$ by $1/\vert\wtd y^{\ast} \wtd x\vert$ instead. Notice that $c_{\lambda}\ge 1$, with equality if $A$ is Hermitian.

If the approximate left eigenvector $\wtd y$ is not available, bounds directly on $\vert\lambda - \wtd\lambda \vert$ do not exist in general. The equation (7.104) would be the only thing available to explain the accuracy of the computed eigenvalues $\wtd\lambda$ and $\wtd x$.