Notes on Simplification of Matrices

These notes discuss how to “simplify” an (n × n) matrix. From the point of view of the larger context of the course, the goal is to have a way to compute e for a given matrix A, which is useful in solving linear ODEs. This is a preliminary version. If you have any comments or corrections, even very minor ones, please let me know. Let us note that the definition of exponentials of matrices and the fact that they are useful for solving linear ODEs are all discussed in the textbook (§14, 15); we will not repeat them here. Some of the material in these notes can also be found in §17, 25 of the textbook. A lot of the material in these notes are drawn from some notes of Eliashberg for a previous version of this course, which you can find online. Before we proceed, let’s start with a few observations: (1) To compute e, it’s useful to find some B which is similar to A, i.e., B = S−1AS for some S, such that e is easier to compute. This is because (Exercise) e = SeBS−1. (2) Now, in terms of computing exponentials, the easiest matrices are diagonal ones. If Λ = diag(λ1, . . . λn), then (Exercise) e = diag(e1 , . . . , en).