Definition 9.4.1 Let V be a vector space over the field of scalars F. Then N ∈ℒ
The following lemma contains some significant observations about nilpotent transformations.
Lemma 9.4.2 Suppose N^{k}x≠0. Then
Proof: Suppose ∑ _{i=0}^{k}c_{i}N^{i}x = 0 where not all c_{i} = 0. There exists l such that k ≤ l < m and N^{l+1}x = 0 but N^{l}x≠0. Then multiply both sides by N^{l} to conclude that c_{0} = 0. Next multiply both sides by N^{l−1} to conclude that c_{1} = 0 and continue this way to obtain that all the c_{i} = 0.
Next consider the claim that λ^{m} is the minimal polynomial. If p

where the degree of r
For such a nilpotent transformation, let

each of these subspaces in the above direct sum being N invariant. For x one of the x_{k}, consider β_{x} given by

where N^{r}x is in the span of the above vectors. Then by the above lemma, N^{r}x = 0.
By Theorem 9.2.5, the matrix of N with respect to the above basis is the block diagonal matrix

where M^{k} denotes the matrix of N restricted to span

Also the cyclic sets β_{x1},β_{x2},


Thus the matrix M_{k} is the r_{k} × r_{k} matrix which has ones down the super diagonal and zeros elsewhere. The following convenient notation will be used.
Definition 9.4.3 J_{k}

In words, there is an unbroken string of ones down the super diagonal and the number α filling every space on the main diagonal with zeros everywhere else.
Then with this definition and the above discussion, the following proposition has been proved.

for some m ∈ ℕ. Here W is a p dimensional vector space with field of scalars F. Then there exists a basis for W such that the matrix of N with respect to this basis is of the form
 (9.5) 
where r_{1} ≥ r_{2} ≥
Observation 9.4.5 Observe that J_{r}
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
Proof: Since J and J^{′} are similar, it follows that for each k an integer, J^{k} and J^{′k} are similar. Hence, for each k, these matrices have the same rank. Now suppose J≠J^{′}. Note first that

Denote the blocks of J as J_{rk}

where M_{rj} = M_{rj′} for j ≤ k − 1 but M_{rk′} is a zero r_{k}^{′}×r_{k}^{′} matrix while M_{rk} is a larger matrix which is not equal to 0. For example, M_{rk} could look like

Thus there are more pivot columns in J^{rk−1} than in