Residual Vector.

Let $\wtd\lambda$ be a computed eigenvalue, and let $\wtd x$ be its corresponding computed eigenvector. For the simplicity of the presentation, we normalize the computed eigenvector so that $\Vert\wtd x\Vert _2 = 1$. The corresponding residual vector or residual error is defined by

$$r = A\wtd x - \wtd\lambda B \wtd x.$$

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