Explore the derivation of eigenvalues and eigenvectors for a matrix "A" to solve the matrix system of equations dx/dt = A*x in this 24-minute educational video. Begin with an overview and problem setup before delving into the simplest system of ODEs with decoupled diagonal dynamics. Discover how to find a coordinate system where dynamics appear diagonal, and follow the step-by-step derivation of the eigenvalue/eigenvector equation. Visualize the eigenvalue equation through helpful illustrations, and conclude with a comprehensive recap and summary. Gain valuable insights into differential equations and matrix systems from Steve Brunton, an expert in the field.