Residual Vectors.

Let $\wtd\lambda$ denote a computed eigenvalue of $A$, and let $\wtd x$ be its corresponding computed eigenvector, i.e.,

$$A \wtd x\approx \wtd\lambda\wtd x.$$

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

$$\wtd y^{\ast} A\approx \wtd\lambda\wtd y^{\ast}.$$

For simplicity, we normalize the computed eigenvectors so that $\Vert\wtd x\Vert _2 = 1$ and $\Vert\wtd y\Vert _2 = 1$. The residual vectors corresponding to the computed eigenvalue $\wtd\lambda$ and right and left computed eigenvectors $\wtd x$ and $\wtd y$ are defined as

$$r = A\wtd x - \wtd\lambda \wtd x \quad \mbox{and} \quad s^{\ast} = \wtd y^{\ast} A-\wtd\lambda\wtd y^{\ast},$$

respectively. We are interested in knowing the accuracy of the computed eigenvalue $\wtd\lambda$ and eigenvectors $\wtd x$ and $\wtd y$ in terms of these residuals.