Residual Vectors.

Let $(\wtd\alpha,\wtd\beta)$ denote a computed eigenvalue, and let $\wtd x$ be its corresponding computed eigenvector, i.e.,

$$\wtd\beta A \wtd x\approx \wtd\alpha B\wtd x.$$

Sometimes, its corresponding left computed eigenvector $\wtd y$ is also available:

$$\wtd\beta \wtd y^{\ast} A\approx \wtd\alpha\wtd y^{\ast} B.$$

The residual vectors corresponding to the computed eigenvalue $(\wtd\alpha,\wtd\beta)$ are defined as

$$r = \wtd\beta A \wtd x - \wtd\alpha B\wtd x \quad \mbox{and}... ...ast} = \wtd\beta \wtd y^{\ast} A - \wtd\alpha\wtd y^{\ast} B,$$

respectively. For the simplicity, we normalize the approximate eigenvectors so that $\Vert\wtd x\Vert _2 = 1$ and $\Vert\wtd y\Vert _2 = 1$ and normalize the approximate eigenvalue so that $\vert\wtd\alpha\vert^2+\vert\wtd\beta\vert^2=1$.