Page 379 - A Course in Linear Algebra with Applications

9.4: Jordan Normal Form 363

respectively, where m; is the sum of the numbers of columns
in Jordan blocks with eigenvalue ci and ki is the number of
columns in the largest such Jordan block.

Example 9.4.9
Find the minimum polynomial of the matrix

A =

The Jordan form of A is

/ 2 1 I 0 \
0 2 1 0
N
— I
V 0 0 1 2 /

by Example 9.4.7. Here 2 is the only eigenvalue and there are
two Jordan blocks, with 2 and 1 columns. The minimum poly-
2
nomial of A is therefore (x — 2) . Of course the characteristic
3
polynomial is (2 — x) ,
Application of Jordan form to differential equations
In 8.3 we studied systems of first order linear differential
equations for functions yi, y.2,..., y n of a variable x. Such a
system takes the matrix form

Y' = AY.

Here Y is the column of functions y\,. .., y n and A is an n x
n matrix with constant coefficients. Since any such matrix
A is similar to a triangular matrix (by 8.1.8), it is possible
to change to a system of linear differential equations for a
new set of functions which has a triangular coefficient matrix.
This new system can then be solved by back substitution,