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 Rôle 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.