We now combine the generalized eigenspace decomposition with the structure theory of nilpotent operators to obtain the Jordan normal form—the “best possible” matrix representation for an operator over an algebraically closed field.
7.7.1 Jordan Blocks
Definition 7.7.1 (Jordan block). A Jordan block of size with eigenvalue is the matrix This has on the diagonal, 1’s on the superdiagonal, and 0’s elsewhere.
Example 7.7.1 (Small Jordan blocks). A Jordan block is just a scalar—this corresponds to a true eigenvector. Larger blocks correspond to generalized eigenvectors that are not true eigenvectors.
Proposition 7.7.1 (Properties of Jordan blocks). Let . Then:
- The only eigenvalue of is , with algebraic multiplicity .
- The eigenspace has dimension 1.
- The generalized eigenspace is all of .
- is nilpotent with index .
Proof. (1) The characteristic polynomial is .
(2) has kernel spanned by (the first standard basis vector).
(3) Since , every vector is in .
(4) (it maps ) but . ∎
Note. A Jordan block represents the action of an operator on a single chain of generalized eigenvectors. The chain is where for and . Equivalently, for and .
Theorem 7.7.1 (Jordan normal form). Let be a finite-dimensional vector space over an algebraically closed field , and let . Then there exists a basis of with respect to which the matrix of is block diagonal: where each block is a Jordan block. This matrix is called the Jordan normal form (or Jordan canonical form) of .
Proof. By the generalized eigenspace decomposition, where are the distinct eigenvalues. Each is -invariant, so we can work on each generalized eigenspace separately.
On , the operator is nilpotent (this was proved earlier). By the structure theorem for nilpotent operators, there exists a chain basis of such that has matrix .
Since , the matrix of in this basis is
Combining the bases for all generalized eigenspaces gives a basis of in which has the desired block diagonal form. ∎
Note. The Jordan form groups blocks by eigenvalue: all blocks for a given appear together, corresponding to the decomposition of into chains. The total size of blocks with eigenvalue equals = algebraic multiplicity of .
7.7.3 Existence and Uniqueness
Theorem 7.7.2 (Uniqueness of Jordan normal form). The Jordan normal form of is unique up to permutation of the blocks. That is, the multiset of Jordan blocks is uniquely determined by .
Proof. For each eigenvalue , the Jordan blocks with eigenvalue correspond to the partition of given by the nilpotent operator . We showed this partition is uniquely determined by dimensions of iterated kernels:
The number of blocks of size exactly is
Since these dimensions are intrinsic to , the block structure is uniquely determined. ∎
Corollary 7.7.1 (Similarity criterion). Two operators are similar if and only if they have the same Jordan normal form (up to block ordering). Equivalently, iff for each and each :
Note. This gives a complete set of similarity invariants. Unlike trace and determinant alone (which don’t distinguish all similarity classes), the full sequence of kernel dimensions completely determines the similarity class.
Given a matrix , here is the procedure to find its Jordan form:
Step 1: Find eigenvalues. Compute and factor it as . The are eigenvalues with algebraic multiplicities .
Step 2: For each eigenvalue , determine block sizes. Compute for until stabilization. The number of blocks of size is . Alternatively:
- Number of blocks = = geometric multiplicity.
- Number of blocks of size is .
- Number of blocks of size exactly is .
Step 3: Assemble the Jordan form. List all Jordan blocks grouped by eigenvalue.
Example 7.7.2 (Computing Jordan form). Let .
Step 1: , so with multiplicity 3.
Step 2: So , , .
Number of blocks = . Size of the single block = 3.
Step 3: Jordan form is . (The matrix is already in Jordan form!)
Example 7.7.3 (A more interesting example). Let .
Step 1: , so with multiplicity 4.
Step 2: , so . , so .
Number of blocks = 2. Blocks of size : . So both blocks have size 2.
Step 3: Jordan form is .
Example 7.7.4 (Multiple eigenvalues). Let .
Step 1: . Eigenvalues: , , each with multiplicity 2.
Step 2 for : Restricted to generalized eigenspace (first two coordinates): , . One block of size 2: .
Step 2 for : Restricted to generalized eigenspace (last two coordinates): . Two blocks of size 1: .
Step 3: Jordan form is .
Note. The last example shows that contributes two blocks because the eigenspace has dimension 2 (geometric = algebraic multiplicity). This is the diagonalizable case for that eigenvalue. In contrast, has a block because geometric multiplicity (1) is less than algebraic multiplicity (2).