Difficulty: Advanced | Prerequisites: Eigenvalues and eigenvectors, characteristic polynomial, diagonalisation, invariant subspaces, nullspace, Spectral Theorem (first half of this lecture).
When an operator cannot be diagonalised (because eigenspaces are too small), the next best thing is the Jordan Canonical Form (JCF). This gives the "almost diagonal" matrix representation that every linear operator over an algebraically closed field admits. Where the Spectral Theorem handles the nice case (self-adjoint, fully diagonalisable), JCF handles the general case. The key new object is the generalized eigenspace, which enlarges ordinary eigenspaces to account for the "missing" dimensions. This material sits at the end of most abstract linear algebra courses and ties together characteristic polynomials, invariant subspaces, and nilpotent operators.
Every linear operator whose characteristic polynomial splits into linear factors (always true over ℂ) has a basis in which its matrix is block-diagonal with Jordan blocks on the diagonal. Each Jordan block has a single eigenvalue on its diagonal and 1s on the superdiagonal. The invariant subspaces corresponding to these blocks are generalized eigenspaces.
Jordan Canonical Form (JCF)
A block-diagonal matrix representation [T]_β = diag(J₁, J₂, …, Jₖ), where each Jᵢ is a Jordan block. In simple terms, this is the simplest matrix shape you can achieve for an operator that is not diagonalisable.
Jordan block
A square matrix of the form Jᵢⱼ with eigenvalue λᵢ on every diagonal entry, 1s on the superdiagonal, and 0s elsewhere. Think of it as "almost a diagonal matrix" with tiny off-diagonal 1s encoding the failure of diagonalisability.
For a Jordan block of size s:
Jᵢⱼ = [ λᵢ 1 0 ··· 0 ]
[ 0 λᵢ 1 ··· 0 ]
[ · · · ··· · ]
[ 0 0 0 ··· λᵢ]
Generalized eigenspace
For an eigenvalue λ of T, the generalized eigenspace is:
K_λ = { v ∈ V | (T − λI)^p v = 0_V for some p ≥ 1 }
In simple terms, this is the set of vectors that are eventually sent to zero by repeated application of (T − λI). Ordinary eigenvectors satisfy (T − λI)v = 0 on the first application; generalized eigenvectors may need two or more applications.
Eigenspace (ordinary)
E_λ = N(T − λI) = { v ∈ V | (T − λI)v = 0_V }
This is the special case of the generalized eigenspace with p = 1. Every ordinary eigenvector is a generalized eigenvector, but not vice versa.
T-invariant subspace
A subspace W with T(W) ⊆ W. Each block in a Jordan form matrix corresponds to a T-invariant subspace.
Algebraic multiplicity
The exponent nⱼ of (λⱼ − t) in the characteristic polynomial det([T] − tIₙ) = (λ₁ − t)^{n₁} ··· (λₖ − t)^{nₖ}. This counts how many times the eigenvalue appears as a root.
Geometric multiplicity
dim(E_λ) = dim N(T − λI). This counts the number of linearly independent eigenvectors for λ. It equals the number of Jordan blocks associated with λ.
The characteristic polynomial must split into linear factors:
det([T] − tIₙ) = (λ₁ − t)^{n₁} ··· (λₖ − t)^{nₖ}
If nⱼ = dim(E_{λⱼ}) for each j = 1, …, k (algebraic multiplicity equals geometric multiplicity), then we can find a basis β such that [T]_β is diagonal:
[T]_β = diag(λ₁^{(n₁)}, λ₂^{(n₂)}, …, λₖ^{(nₖ)})
where λⱼ^{(nⱼ)} means λⱼ repeated nⱼ times.
When this fails (some eigenspace is too small), we need JCF instead.
There exists a basis β of V such that:
[T]_β = diag(J₁, J₂, …, Jₖ)
Each Jordan block Jᵢ is itself block-diagonal, composed of Jordan boxes:
Jᵢ = diag(Jᵢ₁, Jᵢ₂, …, Jᵢₘᵢ)
Each Jordan box Jᵢⱼ is an sᵢⱼ × sᵢⱼ matrix with λᵢ on the diagonal and 1s on the superdiagonal.
Every block in the matrix representation [T]_β corresponds to a T-invariant subspace.
If [T]_β has a block A occupying rows and columns 1 through ℓ, set W = Span{v₁, …, vₗ} (the first ℓ basis vectors). Then T(W) ⊆ W.
This is because T(vⱼ) = a₁ⱼv₁ + … + aₗⱼvₗ for j = 1, …, ℓ, which is a linear combination of vectors in W.
Recall: E_λ = N(T − λI) is the ordinary eigenspace.
The generalized eigenspace is the larger set:
K_λ = { v ∈ V | (T − λI)^p v = 0_V for some p ≥ 1 }
K_λ ⊇ E_λ always. When K_λ is strictly larger than E_λ, the operator has generalized eigenvectors that are not ordinary eigenvectors.
The invariant subspaces for the JCF correspond exactly to the generalized eigenspaces.
(a) K_λ is a T-invariant subspace.
(b) If λ, μ are distinct eigenvalues, then K_λ ∩ K_μ = {0_V}. (Generalized eigenspaces for different eigenvalues intersect trivially.)
(c) (T − μI)|_{K_λ} is bijective (invertible) when μ ≠ λ. In other words, the only eigenvalue that "acts nilpotently" on K_λ is λ itself.
Characteristic polynomial (fully split):
det([T] − tIₙ) = (λ₁ − t)^{n₁}(λ₂ − t)^{n₂} ··· (λₖ − t)^{nₖ}
Generalized eigenspace:
K_λ = { v ∈ V | (T − λI)^p v = 0_V for some p ≥ 1 }
Jordan box (s × s):
[ λ 1 0 ··· 0 ]
[ 0 λ 1 ··· 0 ]
[ · · · ··· · ]
[ 0 0 0 ··· λ ]
Diagonalisability criterion:
T is diagonalisable ⟺ for every eigenvalue λⱼ, algebraic multiplicity nⱼ = geometric multiplicity dim(E_{λⱼ}).
Jordan form appears in the analysis of systems of linear differential equations (dx/dt = Ax), where the structure of Jordan blocks determines whether solutions involve pure exponentials or polynomial-times-exponential terms. It also arises in control theory when analysing the stability and controllability of dynamical systems.
Students often assume that if the characteristic polynomial splits completely, the operator must be diagonalisable. It need not be: the polynomial can split but the eigenspaces can still be too small. JCF is the tool for exactly this situation.
Students confuse generalized eigenvectors with ordinary eigenvectors. An ordinary eigenvector satisfies (T − λI)v = 0; a generalized eigenvector satisfies (T − λI)^p v = 0 for some p ≥ 1. Every ordinary eigenvector is generalized, but a generalized eigenvector with p > 1 is not an ordinary eigenvector.
Students sometimes think the number of Jordan blocks for an eigenvalue λ equals the algebraic multiplicity. It equals the geometric multiplicity (the dimension of the ordinary eigenspace).
Students may think different eigenvalues can share a single Jordan block. They cannot: each block is associated with exactly one eigenvalue.
⚠️ Know the statement of the Jordan Canonical Form theorem and what each piece means (blocks, boxes, eigenvalues on the diagonal, 1s on the superdiagonal).
⚠️ Be able to distinguish when an operator is diagonalisable versus when it requires a non-trivial JCF (algebraic vs geometric multiplicity).
⚠️ You may be asked to define generalized eigenspaces and state their key properties (Theorem 7.1 parts a, b, c).
⚠️ Expect problems asking you to find the JCF of a given matrix or to determine the sizes of Jordan blocks from given information about nullities of powers of (A − λI).
True or False: Every matrix over ℂ has a Jordan Canonical Form.
Fill in the blank: A Jordan box has eigenvalue λ on the diagonal and ______ on the superdiagonal.
True or False: The generalized eigenspace K_λ is always at least as large as the ordinary eigenspace E_λ.
Fill in the blank: An operator is diagonalisable if and only if for every eigenvalue, the algebraic multiplicity equals the ______ multiplicity.
True or False: If λ and μ are distinct eigenvalues, then K_λ ∩ K_μ = {0}.
Q: Define the generalized eigenspace K_λ for an eigenvalue λ of T.
A: K_λ = { v ∈ V | (T − λI)^p v = 0_V for some integer p ≥ 1 }. It contains all vectors eventually annihilated by repeated application of (T − λI).
Q: State the Jordan Canonical Form theorem.
A: For T: Vⁿ → Vⁿ with characteristic polynomial that splits into linear factors, there exists a basis β such that [T]_β is block-diagonal with Jordan blocks. Each Jordan block has a single eigenvalue on its diagonal and 1s on the superdiagonal.
Q: How many Jordan blocks does an eigenvalue λ have, and what determines the sizes?
A: The number of Jordan blocks for λ equals the geometric multiplicity dim(E_λ). Their sizes are determined by the sequence dim N(T − λI)^p for p = 1, 2, 3, … The total size of all blocks for λ equals the algebraic multiplicity.
Q: Explain why every block in a matrix representation [T]_β corresponds to a T-invariant subspace.
A: If a block occupies columns (and rows) 1 through ℓ, set W = Span{v₁, …, vₗ}. For each basis vector vⱼ with j ≤ ℓ, T(vⱼ) is a linear combination of v₁, …, vₗ (because entries outside the block are zero). So T(vⱼ) ∈ W for all j, hence T(W) ⊆ W.
Q: If A is a 4×4 matrix with characteristic polynomial (2 − t)⁴ and dim E₂ = 2, what are the possible Jordan forms?
A: There are two Jordan blocks (since dim E₂ = 2). The block sizes must sum to 4. The possible partitions of 4 into 2 parts are (3,1) and (2,2). So the JCF is either diag(J₃(2), J₁(2)) or diag(J₂(2), J₂(2)), where Jₛ(2) is an s × s Jordan block with eigenvalue 2. Further information (e.g. dim N(A − 2I)²) is needed to distinguish the two.
This connects directly to the Spectral Theorem covered in the first half of this lecture: when T is self-adjoint, the JCF reduces to a diagonal matrix. Jordan form is also closely related to the theory of nilpotent operators (since each block T|_{K_λ} − λI is nilpotent on K_λ). In differential equations, the structure of Jordan blocks determines the form of solutions to constant-coefficient linear systems.
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