Nilpotent operators are the building blocks for understanding non-diagonalizable maps. On each generalized eigenspace , the operator is nilpotent, so understanding nilpotent operators is the key to understanding all operators.
7.6.1 Definition and Basic Properties
Definition 7.6.1 (Nilpotent operator). An operator is nilpotent if for some positive integer . The smallest such is called the index of nilpotency (or nilpotence index) of .
Example 7.6.1 (Basic examples of nilpotent operators).
- The zero operator is nilpotent with index 1.
- The shift operator on with matrix is nilpotent with index .
- Any strictly upper-triangular matrix (zeros on and below the diagonal) is nilpotent.
Proposition 7.6.1 (Characterizations of nilpotent operators). For with , the following are equivalent:
- is nilpotent.
- .
- The only eigenvalue of is (if is algebraically closed).
- .
Proof. (1) (2): If , then the chain stabilizes by step , hence by step (since dimensions are bounded by ). So . But means , so , which means .
(2) (3): Suppose is an eigenvalue with eigenvector . Then , so . If , then , so , hence .
(3) (4): Over an algebraically closed field, by the generalized eigenspace decomposition, . If the only eigenvalue is , then .
(4) (1): If , then . ∎
Note. Over a non-algebraically closed field, (3) needs modification: is nilpotent iff has no eigenvalues in any field extension, or equivalently, iff the characteristic polynomial is .
Corollary 7.6.1 (Nilpotent matrices are similar to strictly upper-triangular matrices). If is nilpotent, then there exists a basis in which the matrix of is strictly upper-triangular.
Proof. We prove this by induction on . If , then and we’re done.
For : since is nilpotent and nonzero (otherwise trivial), . Pick and extend to a basis . The matrix of has first column all zeros.
Alternatively, consider . Since is nilpotent and nonzero, (otherwise would be surjective, hence bijective, contradicting nilpotency). The restriction is well-defined ( is -invariant) and nilpotent. By induction, choose a basis of making strictly upper-triangular, extend to a basis of , and the result follows. ∎
7.6.2 The Nilpotent Basis Construction
The key to understanding nilpotent operators is building a basis adapted to the “chains” of vectors connected by .
Definition 7.6.2 (Chains and Jordan chains). Let be nilpotent. A chain (or Jordan chain) of length is a sequence of vectors such that: The vector is called the head of the chain, and is the tail.
Note. In a chain, , , and generally . The chain encodes how “moves” vectors toward the kernel.
Lemma 7.6.1 (Key lemma for chain construction). Let be nilpotent. Suppose is a subspace satisfying . Then:
- is injective.
- .
- (if ).
Proof. (1) If and , then .
(2) For , we have , so , meaning .
(3) If , then , so . By assumption , so , hence . ∎
Theorem 7.6.1 (Structure theorem for nilpotent operators). Let be nilpotent with index (so but ). Then there exists a basis of of the form where each sequence is a chain: The chain lengths satisfy and .
Proof. We construct the basis by working down from .
Step 1: Let be a complement of in . Pick a basis of . These will be the heads of chains of length .
By the lemma, and . Define for each . These vectors are linearly independent (since is injective).
Step 2: Now work in . We have with . Extend to a larger subspace that complements in . Pick additional basis vectors to span the complement of in . These start new chains of length .
Continue inductively: At each stage, we have chains of various lengths. Applying extends existing chains, and we add new chain heads as needed to complement in .
Termination: At the final step, we’re in , and all chains terminate. The collection of all chain vectors forms a basis of . ∎
Note. The chain lengths form a partition of . This partition is an invariant of the nilpotent operator—two nilpotent operators are similar iff they have the same partition.
7.6.3 Jordan Blocks for Nilpotent Operators
The chain basis gives the matrix of a nilpotent operator a very specific form.
Definition 7.6.3 (Nilpotent Jordan block). A nilpotent Jordan block of size is the matrix This has 1’s on the superdiagonal and 0’s elsewhere.
Corollary 7.6.2 (Matrix form of nilpotent operators). With respect to a chain basis ordered as , the matrix of a nilpotent operator is block diagonal:
Proof. Within each chain , we have for and . In matrix form with this basis ordering, this gives exactly for the -th block. The blocks are independent since chains span complementary subspaces. ∎
Example 7.6.2 (Nilpotent operator with partition ). Consider nilpotent with chain lengths . A chain basis might be: where , , , etc.
The matrix is:
Proposition 7.6.2 (Uniqueness of the Jordan form for nilpotent operators). The partition associated to a nilpotent operator is uniquely determined by . Equivalently, two nilpotent operators are similar iff they have the same Jordan block sizes (counting multiplicity).
Proof. The partition can be recovered from dimensions of iterated kernels. Let . Then:
From this sequence of differences, we can reconstruct the partition: the number of blocks of size exactly is . Since the are determined by , so is the partition. ∎
Note. This dimension-counting argument is the key to proving uniqueness of Jordan form in general. The rank of determines how many Jordan blocks for have size .