9.4. NILPOTENT TRANSFORMATIONS 245

Also the cyclic sets βx1, βx2

, · · · , βxqwill be ordered according to length, the length of βxi

being at least as large as the length of βxi+1,∣∣βxk

∣∣ ≡ rk. Then since Nrkxk = 0, it is now

easy to find Mk. Using the procedure mentioned above for determining the matrix of alinear transformation, (

0 Nrk−1xk · · · Nxk

)=

(Nrk−1xk Nrk−2xk · · · xk

)

0 1 0

0 0. . .

......

. . . 1

0 0 · · · 0

Thus the matrixMk is the rk×rk matrix which has ones down the super diagonal and zeroselsewhere. The following convenient notation will be used.

Definition 9.4.3 Jk (α) is a Jordan block if it is a k × k matrix of the form

Jk (α) =

α 1 0

0. . .

. . ....

. . .. . . 1

0 · · · 0 α

In words, there is an unbroken string of ones down the super diagonal and the number αfilling every space on the main diagonal with zeros everywhere else.

Then with this definition and the above discussion, the following proposition has beenproved.

Proposition 9.4.4 Let N ∈ L (W,W ) be nilpotent,

Nm = 0

for some m ∈ N. Here W is a p dimensional vector space with field of scalars F. Then thereexists a basis for W such that the matrix of N with respect to this basis is of the form

J =

Jr1 (0) 0

Jr2 (0). . .

0 Jrs (0)

 (9.5)

where r1 ≥ r2 ≥ · · · ≥ rs ≥ 1 and∑s

i=1 ri = p. In the above, the Jrj (0) is called a Jordanblock of size rj × rj with 0 down the main diagonal.

Observation 9.4.5 Observe that Jr (0)r= 0 but Jr (0)

r−1 ̸= 0.

In fact, the matrix of the above proposition is unique.

Corollary 9.4.6 Let J, J ′ both be matrices of the nilpotent linear transformation N ∈L (W,W ) which are of the form described in Proposition 9.4.4. Then J = J ′. In fact,if the rank of Jk equals the rank of J ′k for all nonnegative integers k, then J = J ′.