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Nearest-Jordan Structure

Now suppose that the block in the Procrustes problem is allowed
to vary with . Moreover, suppose that is in the nearest
staircase form to ; that is,

for fixed block sizes,
where the *-elements are the corresponding matrix elements of and
the blocks are either fixed or determined by some heuristic, e.g.,
taking the average trace of the blocks they replace in . Then
a minimization of
finds the nearest matrix as a particular
Jordan structure, where the structure is determined by the block sizes
and the eigenvalues are .
When the are fixed, we call this the *orbit* problem,
and when the are selected by the heuristic given we call
this the *bundle* problem.
Such a problem can be useful in regularizing the computation of Jordan
structures of matrices with ill-conditioned eigenvalues.

The form of the differential of , surprisingly, is the same as that
of for the Procrustes problem

This is because
for the selected as above for either the orbit or the bundle case, where
.
In contrast, the form of the second
derivatives is a bit more complicated, since now depends on :

where
,
is just short for the Procrustes ( constant)
part of the second derivative, and
, which is
the staircase part of (with trace averages
or zeros on the diagonal depending on whether bundle or orbit).

** Next:** Trace Minimization
** Up:** Sample Problems and Their
** Previous:** The Procrustes Problem
** Contents**
** Index**
Susan Blackford
2000-11-20