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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