# Introduction to Linear Dynamical Systems

- Institution: Stanford University
- Department: Electrical Engineering
- Teacher: Stephen Boyd
- Level: Undergraduate
- Language: English
- Subjects: Engineering
- Source: [Stanford Engineering Everywhere](https://see.stanford.edu/Course/EE263)
- Licence: [CC BY-NC-SA 4.0](http://creativecommons.org/licenses/by-nc-sa/4.0/legalcode) (Non-commercial use only.)
- Syllabus: [at the source](https://see.stanford.edu/materials/lsoeldsee263/Syllabus.pdf)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems.

Topics include: Least-squares aproximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Matrix exponential, stability, and asymptotic behavior. Multi-input multi-output systems, impulse and step matrices; convolution and transfer matrix descriptions. Control, reachability, state transfer, and least-norm inputs. Observability and least-squares state estimation.

Prerequisites: Exposure to linear algebra and matrices (as in Math. 103). You should have seen the following topics: matrices and vectors, (introductory) linear algebra; differential equations, Laplace transform, transfer functions. Exposure to topics such as control systems, circuits, signals and systems, or dynamics is not required, but can increase your appreciation.

## Lectures

1. [Overview Of Linear Dynamical Systems, Why Study Linear Dynamical Systems?, Examples Of Linear Dynamical Systems, Estimation/Filtering Example, Linear Functions And Examples](https://invisiblenetwork.ai/learn/stanford/ee263/0885faa18b26), 1 h 17 min (Video)
2. [Linear Functions (Continued), Interpretations Of Y=Ax, Linear Elastic Structure, Example, Total Force/Torque On Rigid Body Example, Linear Static Circuit Example, Illumination With Multiple Lamps Example, Cost Of Production Example,…](https://invisiblenetwork.ai/learn/stanford/ee263/53c6f14aca62), 1 h 6 min (Video)
3. [Linearization (Continued), Navigation By Range Measurement, Broad Categories Of Applications, Matrix Multiplication As Mixture Of Columns, Block Diagram Representation, Linear Algebra Review, Basis And Dimension, Nullspace Of A Matrix](https://invisiblenetwork.ai/learn/stanford/ee263/7b8a50e32a2e), 1 h 19 min (Video)
4. [Nullspace Of A Matrix(Continued), Range Of A Matrix, Inverse, Rank Of A Matrix, Conservation Of Dimension, 'Coding' Interpretation Of Rank, Application: Fast Matrix-Vector Multiplication, Change Of Coordinates, (Euclidian) Norm, Inner…](https://invisiblenetwork.ai/learn/stanford/ee263/bfe92297f7db), 1 h 14 min (Video)
5. [Orthonormal Set Of Vectors, Geometric Interpretation, Gram-Schmidt Procedure, General Gram-Schmidt Procedure, Applications Of Gram-Schmidt Procedure, 'Full' QR Factorization, Orthogonal Decomposition Induced By A, Least-Squares](https://invisiblenetwork.ai/learn/stanford/ee263/8ce57faad37f), 1 h 15 min (Video)
6. [Least-Squares, Geometric Interpretation, Least-Squares (Approximate) Solution, Projection On R(A), Least-Squares Via QR Factorization, Least-Squares Estimation, Blue Property, Navigation From Range Measurements, Least-Squares Data Fitting](https://invisiblenetwork.ai/learn/stanford/ee263/dc0a660c9002), 1 h 16 min (Video)
7. [Least-Squares Polynomial Fitting, Norm Of Optimal Residual Versus P, Least-Squares System Identification, Model Order Selection, Cross-Validation, Recursive Least-Squares, Multi-Objective Least-Squares](https://invisiblenetwork.ai/learn/stanford/ee263/6577db12afd3), 1 h 16 min (Video)
8. [Multi-Objective Least-Squares, Weighted-Sum Objective, Minimizing Weighted-Sum Objective, Regularized Least-Squares, Laplacian Regularization, Nonlinear Least-Squares (NLLS), Gauss-Newton Method, Gauss-Newton Example, Least-Norm Solutions…](https://invisiblenetwork.ai/learn/stanford/ee263/d463573a091f), 1 h 16 min (Video)
9. [Least-Norm Solution, Least-Norm Solution Via QR Factorization, Derivation Via Langrange Multipliers, Example: Transferring Mass Unit Distance, Relation To Regularized Least-Squares, General Norm Minimization With Equality Constraints,…](https://invisiblenetwork.ai/learn/stanford/ee263/13cd94d5c0d3), 1 h 9 min (Video)
10. [Examples Of Autonomous Linear Dynamical Systems, Finite-State Discrete-Time Markov Chain, Numerical Integration Of Continuous System, High Order Linear Dynamical Systems, Mechanical Systems, Linearization Near Equilibrium Point,…](https://invisiblenetwork.ai/learn/stanford/ee263/7ecda0fec499), 1 h 12 min (Video)
11. [Solution Via Laplace Transform And Matrix Exponential, Laplace Transform Solution Of X_^ = Ax, Harmonic Oscillator Example, Double Integrator Example, Characteristic Polynomial, Eigenvalues Of A And Poles Of Resolvent, Matrix Exponential,…](https://invisiblenetwork.ai/learn/stanford/ee263/df853dd3b322), 1 h 9 min (Video)
12. [Time Transfer Property, Piecewise Constant System, Qualitative Behavior Of X(T), Stability, Eigenvectors And Diagonalization, Scaling Interpretation, Dynamic Interpretation, Invariant Sets, Summary, Markov Chain (Example)](https://invisiblenetwork.ai/learn/stanford/ee263/d85e68eecb81), 1 h 14 min (Video)
13. [Markov Chain (Example), Diagonalization, Distinct Eigenvalues, Digaonalization And Left Eigenvectors, Modal Form, Diagonalization Examples, Stability Of Discrete-Time Systems, Jordan Canonical Form, Generalized Eigenvectors](https://invisiblenetwork.ai/learn/stanford/ee263/28f1a17307c5), 1 h 13 min (Video)
14. [Jordan Canonical Form, Generalized Modes, Cayley-Hamilton Theorem, Proof Of C-H Theorem, Linear Dynamical Systems With Inputs & Outputs, Block Diagram, Transfer Matrix, Impulse Matrix, Step Matrix](https://invisiblenetwork.ai/learn/stanford/ee263/aa0432045dcc), 1 h 18 min (Video)
15. [DC Or Static Gain Matrix, Discretization With Piecewise Constant Inputs, Causality, Idea Of State, Change Of Coordinates, Z-Transform, Symmetric Matrices, Quadratic Forms, Matrix Nom, And SVD, Eigenvalues Of Symmetric Matrices,…](https://invisiblenetwork.ai/learn/stanford/ee263/fc9a4ed79d24), 1 h 9 min (Video)
16. [RC Circuit (Example), Quadratic Forms, Examples Of Quadratic Form, Inequalities For Quadratic Forms, Positive Semidefinite And Positive Definite Matrices, Matrix Inequalities, Ellipsoids, Gain Of A Matrix In A Direction, Matrix Norm,…](https://invisiblenetwork.ai/learn/stanford/ee263/5b75be5fc927), 1 h 13 min (Video)
17. [Gain Of A Matrix In A Direction, Singular Value Decomposition, Interpretations, Singular Value Decomposition (SVD) Applications, General Pseudo-Inverse, Pseudo-Inverse Via Regularization, Full SVD, Image Of Unit Ball Under Linear…](https://invisiblenetwork.ai/learn/stanford/ee263/7e57cdfa0cf6), 1 h 17 min (Video)
18. [Sensitivity Of Linear Equations To Data Error, Low Rank Approximations, Distance To Singularity, Application: Model Simplification, Controllability And State Transfer, State Transfer, Reachability, Reachability For Discrete-Time LDS](https://invisiblenetwork.ai/learn/stanford/ee263/a9433eae8c64), 1 h 15 min (Video)
19. [Reachability, Controllable System, Lest-Norm Input For Reachability, Minimum Energy Over Infinite Horizon, Continuous-Time Reachability, Impulsive Inputs, Least-Norm Input For Reachability](https://invisiblenetwork.ai/learn/stanford/ee263/f7f71469ab8e), 1 h 10 min (Video)
20. [Continuous-Time Reachability, General State Transfer, Observability And State Estimation, State Estimation Set Up, State Estimation Problem, Observability Matrix, Least-Squares Observers, Some Parting Thoughts..., Linear Algebra, Levels…](https://invisiblenetwork.ai/learn/stanford/ee263/3028ab1719fc), 1 h 9 min (Video)