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:
With second derivatives given by