Next: Trace Minimization
Up: Sample Problems and Their
Previous: The Procrustes Problem
  Contents
  Index
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