Scalars, fields, vectors and vector spaces and subspaces. Basis and dimension, linear transformation and mapping, matrices, range and null spaces. Metrics - norms. Inner products. Similarity Transformations, Invariance and Invariant Subspaces, Eigenvalues and eigenvectors. Projections - Idempotnece and Nilpotence. Orthogonality and Orthonormal basis, Gram-Schmidt Orthonormalization
Linear dynamical systems, State Space Representation, Canonical Forms : Jordan, Companion/Controllable Canonical form. Solution to state equation, The Matrix exponent : Interpretation, properties and computation aspects, Controllability and Observability - Necessary and Sufficient Conditions, stability - Internal Stability and Output Stability. Zero Dynamics - Differentiation of Zero Output and Zero State
Controllable and Observable Subspaces and their Complements. The Kalman Decomposition. Pole-Zero Cancellation, Minimal Representation.
State Feedback Control Design, Stabilizability and Pole Placement. Observer Design and Detectibility. Separation Principle. Reduced Order Observers. Functional Observers and Observer Design for Control
Linear Quadratic Regulator : The Algebraic Ricatti Equation