Residual Vector.

Let $(\wtd\alpha,\wtd\beta)$ denote a computed eigenvalue, and let $\wtd x$ be its corresponding computed eigenvector. For the simplicity, we normalize the computed eigenvector and eigenvalue so that

$$\Vert\wtd x\Vert _2 = 1, \quad \vert\wtd\alpha\vert^2+\vert\wtd\beta\vert^2=1.$$

The corresponding residual vector or residual error is defined by

$$r = \wtd\beta A\wtd x - \wtd\alpha B\wtd x.$$

Ideally, we would like to have $r=0$, but in practice $\Vert r\Vert _2$ is small. It is conceivable that a small residual error implies good accuracy in the computed $(\wtd\alpha,\wtd\beta)$ and $\wtd x$. We are interested in knowing how accurate the computed $(\wtd\alpha,\wtd\beta)$ and $\wtd x$ are, given $r$.