246 CHAPTER 9. CANONICAL FORMS

Proof: Since J and J ′ are similar, it follows that for each k an integer, Jk and J ′k aresimilar. Hence, for each k, these matrices have the same rank. Now suppose J ̸= J ′. Notefirst that

Jr (0)r= 0, Jr (0)

r−1 ̸= 0.

Denote the blocks of J as Jrk (0) and the blocks of J ′ as Jr′k (0). Let k be the first such thatJrk (0) ̸= Jr′k (0). Suppose that rk > r′k. By block multiplication and the above observation,

it follows that the two matrices Jrk−1 and J ′rk−1 are respectively of the forms

Mr1 0. . .

Mrk

∗. . .

0 ∗

,



Mr′10

. . .

Mr′k

0. . .

0 0

whereMrj =Mr′j

for j ≤ k−1 butMr′kis a zero r′k×r′k matrix whileMrk is a larger matrix

which is not equal to 0. For example, Mrk could look like

Mrk =

0 · · · 1

. . ....

0 0

Thus there are more pivot columns in Jrk−1 than in (J ′)

rk−1, contradicting the requirement

that Jk and J ′k have the same rank. ■

9.5 The Jordan Canonical Form

The Jordan canonical form has to do with the case where the minimal polynomial of A ∈L (V, V ) splits. Thus there exist λk in the field of scalars such that the minimal polynomialof A is of the form

p (λ) =

r∏k=1

(λ− λk)mk

Recall the following which follows from Theorem 8.4.4.

Proposition 9.5.1 Let the minimal polynomial of A ∈ L (V, V ) be given by

p (λ) =

r∏k=1

(λ− λk)mk

Then the eigenvalues of A are {λ1, · · · , λr}.

It follows from Corollary 9.2.3 that

V = ker (A− λ1I)m1 ⊕ · · · ⊕ ker (A− λrI)

mr

≡ V1 ⊕ · · · ⊕ Vr

where I denotes the identity linear transformation. Without loss of generality, let thedimensions of the Vk be decreasing from left to right. These Vk are called the generalizedeigenspaces.