Vol. 10 No 6 - Pi Mu Epsilon

464 PI MU EPSILON JOURNAL
[1, page 109] of solving the system x'=Ax, where A is ann x n matrix and
x = (x 1 , ••• ,x,.)T, depends heavily on the commutativity of matrix
multiplication of two appropriate matrices.
Before we go further, we recall two definitions. An n x n matrix Y is
said to be diagonalizable if there exists a nonsingular matrix X and a
diagonal matrix D such that x-lyx. =D. Ann x n matrix Cis said to be
nilpotent if ck = 0 for some positive integer k. Now consider the system
x' = Ax of differential equations, where A is an n x n matrix and x =
(x 1 , ... ,x,.JT. Hirsch & Smale [1, page 109] indicate the following method
of solving this system.
Let A = B + C where B is diagonalizable, C is nilpotent, and BC =
CB. Then the solution is e-4' = e (B+C)t = ,Jlt ec 1 • Here ,/l' and ec' are
easily determined. The following examples illustrate the importance of the
condition BC = CB in solving systems of equations of this form.
X'l = Xz +Xl
Example 1. Consider the system X' 2 = - X 2 + X 3 •
{
x'l =
Let A"~~ -~ -~l • C+Bwhere C" ~~ -~ -~l and B "~~ ~ ~]·
-xl
Here, B 2 = ~~ ~ ~1 and B 3 = ~~ ~ ~]. Thus, B is nilpotent.
000 000
Consider eAt = e