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. This study addresses both classical and previously unsolved optimization tasks in linear algebra, linear systems theory, control theory and signal processing. Exploiting developments in Computation such as Parallel Computing and Neural Networks, it gives a dynamical systems approach for tackling a wide class of constrained optimization tasks. Optimization and Dynamical Systems will be of interest to engineers and mathematicians. Engineers will learn the mathematics and the technical approach necessary to solve a wide class of constrained optimization tasks. Mathematicians will see how techniques from global analysis and differential geometry can be developed to achieve useful construction procedures for optimization on manifolds. Front Matter....Pages i-xiii Matrix Eigenvalue Methods....Pages 1-42 Double Bracket Isospectral Flows....Pages 43-80 Singular Value Decomposition....Pages 81-100 Linear Programming....Pages 101-124 Approximation and Control....Pages 125-161 Balanced Matrix Factorizations....Pages 163-200 Invariant Theory and System Balancing....Pages 201-227 Balancing via Gradient Flows....Pages 229-267 Sensitivity Optimization....Pages 269-309 Back Matter....Pages 311-403 Contents: Matrix Eigenvalue Methods Double Bracket Isospectral Flows Singular Value Decomposition Linear Programming Approximation and Control Balanced Matrix Factorizations Invariant Theory and System Balancing Balancing via Gradient Flows Sensitivity Optimization Linear Algebra Dynamical Systems Global Analysis. This monograph addresses both classical and previously unsolved optimization tasks in algebra, linear systems theory, control theory and signal processing. It exploits a dynamical systems approach for tackling a wide variety of constrained optimization tasks.
