Optimization and Dynamical Systems

1 Matrix Eigenvalue Methods.- 1.1 Introduction.- 1.2 Power Method for Diagonalization.- 1.3 The Rayleigh Quotient Gradient Flow.- 1.4 The QR Algorithm.- 1.5 Singular Value Decomposition (SVD).- 1.6 Standard Least Squares Gradient Flows.- 2 Double Bracket Isospectral Flows.- 2.1 Double Bracket Flows for Diagonalization.- 2.2 Toda Flows and the Riccati Equation.- 2.3 Recursive Lie-Bracket Based Diagonalization.- 3 Singular Value Decomposition.- 3.1 SVD via Double Bracket Flows.- 3.2 A Gradient Flow Approach to SVD.- 4 Linear Programming.- 4.1 The Role of Double Bracket Flows.- 4.2 Interior Point Flows on a Polytope.- 4.3 Recursive Linear Programming/Sorting.- 5 Approximation and Control.- 5.1 Approximations by Lower Rank Matrices.- 5.2 The Polar Decomposition.- 5.3 Output Feedback Control.- 6 Balanced Matrix Factorizations.- 6.1 Introduction.- 6.2 Kempf-Ness Theorem.- 6.3 Global Analysis of Cost Functions.- 6.4 Flows for Balancing Transformations.- 6.5 Flows on the Factors X and Y.- 6.6 Recursive Balancing Matrix Factorizations.- 7 Invariant Theory and System Balancing.- 7.1 Introduction.- 7.2 Plurisubharmonic Functions.- 7.3 The Azad-Loeb Theorem.- 7.4 Application to Balancing.- 7.5 Euclidean Norm Balancing.- 8 Balancing via Gradient Flows.- 8.1 Introduction.- 8.2 Flows on Positive Definite Matrices.- 8.3 Flows for Balancing Transformations.- 8.4 Balancing via Isodynamical Flows.- 8.5 Euclidean Norm Optimal Realizations.- 9 Sensitivity Optimization.- 9.1 A Sensitivity Minimizing Gradient Flow.- 9.2 Related L2-Sensitivity Minimization Flows.- 9.3 Recursive L2-Sensitivity Balancing.- 9.4 L2-Sensitivity Model Reduction.- 9.5 Sensitivity Minimization with Constraints.- A Linear Algebra.- A.1 Matrices and Vectors.- A.2 Addition and Multiplication of Matrices.- A.3 Determinant and Rank of a Matrix.- A.4 Range Space, Kernel and Inverses.- A.5 Powers, Polynomials, Exponentials and Logarithms.- A.6 Eigenvalues, Eigenvectors and Trace.- A.7 Similar Matrices.- A.8 Positive Definite Matrices and Matrix Decompositions.- A.9 Norms of Vectors and Matrices.- A.10 Kronecker Product and Vec.- A.11 Differentiation and Integration.- A.12 Lemma of Lyapunov.- A.13 Vector Spaces and Subspaces.- A.14 Basis and Dimension.- A.15 Mappings and Linear Mappings.- A.16 Inner Products.- B Dynamical Systems.- B.1 Linear Dynamical Systems.- B.2 Linear Dynamical System Matrix Equations.- B.3 Controllability and Stabilizability.- B.4 Observability and Detectability.- B.5 Minimality.- B.6 Markov Parameters and Hankel Matrix.- B.7 Balanced Realizations.- B.8 Vector Fields and Flows.- B.9 Stability Concepts.- B.10 Lyapunov Stability.- C Global Analysis.- C.1 Point Set Topology.- C.2 Advanced Calculus.- C.3 Smooth Manifolds.- C.4 Spheres, Projective Spaces and Grassmannians.- C.5 Tangent Spaces and Tangent Maps.- C.6 Submanifolds.- C.7 Groups, Lie Groups and Lie Algebras.- C.8 Homogeneous Spaces.- C.9 Tangent Bundle.- C.10 Riemannian Metrics and Gradient Flows.- C.11 Stable Manifolds.- C.12 Convergence of Gradient Flows.- References.- Author Index.