Introduction and Autonomous Linear Dynamical Systems

Corresponds to lectures: Overview; Example: Input Design; Example: Prediction; Autonomous linear dynamical systems; Solution via matrix exponential; Dynamic interpretation of eigenvectors; Jordan canonical form. Opens with two motivating examples (designing an input to steer a system, predicting a system’s future state) that justify treating $\dot x = Ax$ as an object worth studying in its own right, then solves it: the matrix exponential $e^{At}$ generalizes the scalar solution $e^{at}$, and diagonalizing $A$ (when possible) turns this into a sum of decoupled modes, each an eigenvector evolving at a rate set by its eigenvalue — the dynamic meaning of diagonalization. When $A$ isn’t diagonalizable, the Jordan canonical form gives the next-best decomposition, producing polynomial-times-exponential terms instead of pure exponentials.

Outline (TODO — flesh out each)

• Overview — state-space models
• Example: input design
• Example: prediction
• Autonomous linear dynamical systems — $\dot x = Ax$
• Solution via the matrix exponential
• Dynamic interpretation of eigenvectors — modes
• Jordan canonical form — the non-diagonalizable case