Since the eigenproblem for a general matrix can also be written as the generalized eigenproblem , 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 is singular, then has at least one infinite eigenvalue. In order to speak of small perturbations to 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 , finite or infinite, can always be written as a ratio , where . For example, if for some , then and . The chordal metric , where is the acute angle between the two vectors and . We may compute this as or if neither nor is infinite. For example, If is infinite and is very large, then is small, so that and are close.
Another, geometric way to understand the chordal metric is with the Riemann sphere : 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 in the complex plane, draw a straight line connecting to the north pole of the Riemann sphere, and let be the point where the straight line and the Riemann sphere intersect. Then is just the Euclidean distance between the two points on the and on the Riemann sphere.
The chordal metric is used to measure changes in eigenvalues in §8.8. For example, the single infinite eigenvalue of is well-conditioned, because changing to can change the infinite eigenvalue to , and .
We refer to §8.8 for further details.