Conditioning

Since the eigenproblem for a general matrix $A$ can also be written as the generalized eigenproblem $A - \lambda I$, all the comments about conditioning in §2.5.5 apply here: eigenvalues can be well-conditioned or ill-conditioned and small perturbations can even make eigenvectors disappear entirely.

When $B$ is singular, then $A - \lambda B$ has at least one infinite eigenvalue. In order to speak of small perturbations to $B$ causing ``small perturbations'' to an infinite eigenvalue, we need an appropriate definition. This is supplied by the chordal metric, which we describe in several equivalent ways. First, any eigenvalue $\lambda_i$, finite or infinite, can always be written as a ratio $\alpha_i / \beta_i$, where $\vert\alpha_i\vert^2 + \vert\beta_i\vert^2=1$. For example, if $\vert\lambda_i\vert = \tan \theta$ for some $\theta$, then $\vert\alpha_i\vert = \sin \theta$ and $\vert\beta_i\vert = \cos \theta$. The chordal metric $\chi ( \lambda_1 , \lambda_2 ) = \sin \tau$, where $\tau$ is the acute angle between the two vectors $(\alpha_1 , \beta_1)$ and $(\alpha_2 , \beta_2)$. We may compute this as $\chi ( \lambda_1 , \lambda_2 ) = \vert \alpha_1 \beta_2 - \alpha_2 \beta_1 \vert$ or $\chi ( \lambda_1 , \lambda_2 ) = \frac{\vert \lambda_1 - \lambda_2 \vert} {\sqrt{1 + \vert\lambda_1\vert^2} \sqrt{1 + \vert\lambda_2\vert^2}}$ if neither $\lambda_1$ nor $\lambda_2$ is infinite. For example, If $\lambda_1 = 1/0$ is infinite and $\lambda_2 = \sqrt{1-10^{-16}}/10^{-8} \approx 10^8$ is very large, then $\chi ( \lambda_1 , \lambda_2 ) = 10^{-8}$ is small, so that $\lambda_1$ and $\lambda_2$ are close.

Another, geometric way to understand the chordal metric is with the Riemann sphere [4]: Imagine a three-dimensional ball of radius 1 with center at the origin of the complex plane; the ball's surface is the Riemann sphere. For any $\lambda$ in the complex plane, draw a straight line connecting $\lambda$ to the north pole of the Riemann sphere, and let $\hat{\lambda}$ be the point where the straight line and the Riemann sphere intersect. Then $\chi ( \lambda_1 , \lambda_2 )$ is just the Euclidean distance $\Vert \hat{\lambda}_1 - \hat{\lambda}_2 \Vert _2$ between the two points on the $\hat{\lambda}_1$ and $\hat{\lambda}_2$ on the Riemann sphere.

The chordal metric is used to measure changes in eigenvalues in §8.8. For example, the single infinite eigenvalue of $1 - \lambda \cdot 0$ is well-conditioned, because changing $0$ to $10^{-8}$ can change the infinite eigenvalue to $10^8$, and $\chi ( 10^8 , \infty ) \approx 10^{-8}$.

We refer to §8.8 for further details.