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###

Simultaneous Schur Decomposition Problem

Consider two matrices and which in the absence of error have
the same Schur vectors; i.e., there is a
such
that and are both block upper triangular
where is the set of by orthogonal matrices.
Now suppose that and are somewhat noisy from measurement errors or
some other kind of lossy filtering.
In that case the that upper triangularizes might not
upper triangularize as well. How does one find the best ?

This is a problem that was presented to us by Schilders
[396], who phrased it as a least squares minimization of
, where
is a mask returning the block lower triangular part of ,
where is broken up into blocks.

For this problem the differential is a bit tricky and its derivation
instructive:

where the second equation results from observing that
and
the third from properties of the trace.
With second derivatives given by

where
,
and
.

** Next:** Simultaneous Diagonalization
** Up:** Sample Problems and Their
** Previous:** Trace Minimization with a
** Contents**
** Index**
Susan Blackford
2000-11-20