Ramblings about the structure of an operator in the presence of an
orthogonal/hermitian structure

1: qua operator
  Given $A$, think of it as an operator;
  then by structure theory of an operator (Jordan),
  we know that $V$ decomposes as generalized eigenspaces of $A$.
  a: each gen.eigenspace has orthonormal bases;
    however, the different gen.eigenspaces might have some angle
    between them -- this is an intrinsic invariant
  b: on the gen.eigenspaces, there's the flags of images/kernels
    of $(A-\lambda I)^k$; these have subordinate ortho bases,
    but I dunno how Jordan blocks relate to these

2: qua form
$[A,A^*]=AA^* - A^*A$ is invariant under $U(V)$
($O(V)$ for transpose);
Sylvester shows that this vanishes iff you're diagonalizable,
and then your singular values are uniquely determined up to order.
More generally, you can upper-triangulize $A$ essentially uniquely
(I think) up to $W$ (Weyl group) [this is essentially getting
$[A,A^*]$ into the skew-hermite/sym matrices, i.e., $\su_n/\so_n$].
So here the classification is almost identical, and not too bad.

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Essentially, you get some natural invariants, and in general the
classification problem is ugly.
