(DiffEQ, Fall 23) 21 - Matrix exponentials

In this lecture we introduce the fundamental matrix, see how it connects to matrix exponentials and how to use this to find solutions. We also look at what happens in the 2x2 case for repeated eigenvectors giving an alternative to what was done in the previous lecture. More information: https://www.calc4.org/flipped-fall-2023 Presenter: https://www.stevebutler.org/ 0:00 Introduction to idea of matrix exponentials 4:49 Fundamental matrices; exponential matrix 14:39 Example (fundamental matrix) 27:08 Computing matrix exponential as series 29:44 Example (exponential of diagonal matrix) 35:15 Making matrices look diagonal 38:19 Computing matrix exponentials 42:32 Repeated eigenvalue case 45:43 Example (exponential matrix; repeated) 48:17 Example (repeated eigenvalues)