This work is aimed at mathematics and engineering graduate students and
researchers in the areas of optimization, dynamical systems, control
sys- tems, signal processing, and linear algebra. The motivation for the
results developed here arises from advanced engineering applications and
the emer- gence of highly parallel computing machines for tackling such
applications. The problems solved are those of linear algebra and linear
systems the- ory, and include such topics as diagonalizing a symmetric
matrix, singular value decomposition, balanced realizations, linear
programming, sensitivity minimization, and eigenvalue assignment by
feedback control. The tools are those, not only of linear algebra and
systems theory, but also of differential geometry. The problems are
solved via dynamical sys- tems implementation, either in continuous time
or discrete time, which is ideally suited to distributed parallel
processing. The problems tackled are indirectly or directly concerned
with dynamical systems themselves, so there is feedback in that
dynamical systems are used to understand and optimize dynamical systems.
One key to the new research results has been the recent discovery of
rather deep existence and uniqueness results for the solution of certain
matrix least squares optimization problems in geomet- ric invariant
theory. These problems, as well as many other optimization problems
arising in linear algebra and systems theory, do not always admit
solutions which can be found by algebraic methods.
