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##

Conditioning

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* [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 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.

** Next:** Specifying an Eigenproblem
** Up:** Generalized Non-Hermitian Eigenproblems
** Previous:** Eigendecompositions
** Contents**
** Index**
Susan Blackford
2000-11-20