472 APPENDIX A. THE JORDAN CANONICAL FORM*

A and B be {λ 1, · · · ,λ r} . Consider the two sequences of numbers {rank(A−λ I)m} and{rank(B−λ I)m}. Since A and B are similar, these two sequences coincide. (Why?) Also,for the same reason, {rank(JA−λ I)m} coincides with {rank(JB−λ I)m} . Now pick λ k aneigenvalue and consider {rank(JA−λ kI)m} and {rank(JB−λ kI)m} . Then

JA−λ kI =



JA (λ 1−λ k) 0. . .

JA (0). . .

0 JA (λ r−λ k)

and a similar formula holds for JB−λ kI. Here

JA (0) =

Jk1 (0) 0

Jk2 (0). . .

0 Jkr (0)

and

JB (0) =

Jl1 (0) 0

Jl2 (0). . .

0 Jlp (0)

and it suffices to verify that li = ki for all i. As noted above, ∑ki = ∑ li. Now from the aboveformulas,

rank(JA−λ kI)m = ∑i̸=k

mi + rank(JA (0)m)

= ∑i̸=k

mi + rank(JB (0)m)

= rank(JB−λ kI)m ,

which shows rank(JA (0)m) = rank(JB (0)

m) for all m. However,

JB (0)m =

Jl1 (0)

m 0Jl2 (0)

m

. . .

0 Jlp (0)m

with a similar formula holding for JA (0)

mand rank(JB (0)m) = ∑

pi=1 rank

(Jli (0)

m) , similarfor rank(JA (0)

m) . In going from m to m+1,

rank(Jli (0)

m)−1 = rank(

Jli (0)m+1)