Gilbert Strang: Eigenvalues and Eigenvectors (Differential Equations and Linear Algebra video lectures)
by Gilbert Strang · MIT OpenCourseWare (RES.18-009)
Eight short MIT lectures by Gilbert Strang that apply eigenvalues to differential equations. Topics are diagonalization, matrix powers and Markov matrices, solving linear systems du/dt = Au, the matrix exponential, similar and symmetric matrices, and second-order systems.
More resources on Systems of ODEs
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Introduction to Differential Equations
Learn differential equations: explore methods for solving, modeling, and analyzing them. Taught by the Khan Academy Team.
Differential Equations and Linear Algebra
Textbook by Gilbert Strang (Wellesley-Cambridge Press, 2014) that teaches ODEs and linear algebra together. Its eigenvalue chapter covers du/dt = Au, the matrix exponential, and stability. Paired with free MIT video lectures for each section. The intermediate-level counterpart to the more advanced Hirsch/Smale/Devaney.
How (and why) to raise e to the power of a matrix | DE6
A 27-minute animated lecture, chapter 6 of 3Blue1Brown's differential equations series. It defines the matrix exponential through its Taylor series and shows why exp(Mt) solves linear systems x' = Mx, visualized as flow in state space. Best geometric intuition for the topic's central object, the matrix exponential, and its link to state-space flow.
Notes on Diffy Qs, Chapter 3: Systems of ODEs
Free open textbook chapter by Jiří Lebl. Covers matrices and linear systems, the eigenvalue method, 2D vector fields, second-order systems and mass-spring applications, repeated eigenvalues, matrix exponentials, and nonhomogeneous systems. Includes exercises with answers. More rigorous than Paul's Notes on the eigenvalue method and matrix exponential while staying readable.
MIT 18.03SC Differential Equations, Unit IV: First-order Systems
Self-paced MIT unit with lecture videos, recitation videos, notes, and problem sets with solutions. Covers linear systems, eigenvalue and normal-mode methods, phase portraits, matrix exponentials, nonlinear systems, linearization near critical points, limit cycles, and chaos. The strongest free structured course for this topic.