Page 383 - A Course in Linear Algebra with Applications

9.4: Jordan Normal Form 367

We solve this system, beginning with the last equation, and
2x
obtain u 3 = c 2e . Next u' 2 = 2u 2, so that u 2 = c\e 2x . Finally
we solve
2x
u[ - 2ui = cie ,

a first order linear equation, and find the solution to be

2x
u\ — (CQ + c 1x)e .

Therefore

ci I <?\

and since Y = SU, we obtain

Co + C\ +CiX
2x
Y = | -CQ-CXX \ e ,
c 2

from which the values of the functions yi,y2,V3 can be read
off.

Exercises 9.4

1. Find the minimum polynomials of the following matrices
by inspection :

w(SS>w(Si) i ( c , (!S

/ 3 0 0'
(d) 0 2 1
\ 0 0 2
2. Let A be an n x n matrix and S an invertible n x n ma-
trix over a field F. If / is any polynomial over F, show that
1
1
f(S~ AS) = S~ f(A)S. Use this result to give another proof