Generalized Eigenspaces and the Jordan Canonical Form

Every complex square matrix is similar to a Jordan canonical form: a block diagonal matrix whose blocks J_k(λ) have the eigenvalue λ on the diagonal and ones directly above it; the blocks come from chains of generalized eigenvectors, vectors v with (A − λI)^kv = 0 for some k. In brief: A = [[3, 1], [−1, 1]] has the single eigenvalue 2 with only one eigenvector, (1, −1), so it is not diagonalizable, but with the generalized eigenvector w = (1, 0) it becomes the Jordan block [[2, 1], [0, 2]] in the basis (1, −1), (1, 0).

The Jordan form is the best possible simplification of a matrix when diagonalization fails. This guide defines generalized eigenspaces, shows how block sizes are read from the kernels of (A − λI)^j, builds a Jordan basis step by step, lists the possible forms for a given characteristic polynomial and explains applications to powers and differential equations, with worked examples and a table.

What are generalized eigenspaces?

For an eigenvalue λ of A, the generalized eigenspace is G_λ = ker (A − λI)ⁿ, the set of vectors killed by some power of A − λI. It contains the ordinary eigenspace ker (A − λI), and its dimension equals the algebraic multiplicity of λ. Over ℂ the whole space is the direct sum of the generalized eigenspaces for all eigenvalues, and each is invariant under A. The kernels increase: ker (A − λI) ⊆ ker (A − λI)² ⊆ … until they stabilize.

What does a Jordan block look like?

The Jordan block of size k for the eigenvalue λ is the k × k matrix with λ on the diagonal, 1 on the superdiagonal and 0 elsewhere. For k = 3: [[λ, 1, 0], [0, λ, 1], [0, 0, λ]]. A Jordan form is a block diagonal matrix of such blocks. Its diagonal carries the eigenvalues, and a block of size 1 is just the eigenvalue. A matrix is diagonalizable exactly when all its Jordan blocks have size 1.

The Jordan theorem: every n × n matrix over ℂ is similar to a Jordan form, which is unique up to the order of the blocks. So two matrices are similar if and only if they have the same Jordan form.

How do you find the block sizes?

Let d_j = dim ker (A − λI)^j, with d₀ = 0. Then:

• the number of blocks for λ (of any size) is d₁, the geometric multiplicity;
• the number of blocks of size at least j is d_j − d_j−1;
• the size of the largest block is the smallest j with d_j = d_j+1, equal to the multiplicity of λ in the minimal polynomial;
• the sum of all block sizes for λ is the algebraic multiplicity.

Example 1: reading sizes from kernel dimensions

A nilpotent 4 × 4 matrix N (the only eigenvalue is 0) with the kernel dimensions d₁ = 2, d₂ = 3, d₃ = 4. The differences are 2, 1, 1: there are 2 blocks, 1 block of size at least 2, 1 block of size at least 3 and none of size at least 4. So the blocks have sizes 3 and 1, and the minimal polynomial is λ³.

Example 2: a single Jordan chain

N = [[0, 1, 0], [0, 0, 1], [0, 0, 0]] has N² = [[0, 0, 1], [0, 0, 0], [0, 0, 0]] and N³ = O. The kernel dimensions are 1, 2, 3, so there is one block of size 3: N is itself a Jordan block.

How do you build a Jordan basis?

A Jordan chain for a block of size k is a list v₁, …, v_k with (A − λI)v₁ = 0 and (A − λI)v_j = v_j−1. Equivalently Av₁ = λv₁ and Av_j = λv_j + v_j−1, which gives exactly the columns of the Jordan block. Choose a vector v_k in ker (A − λI)^k that is not in ker (A − λI)^k−1 and compute the chain downwards by applying A − λI.

Example 3: a 2 × 2 matrix

A = [[3, 1], [−1, 1]]: tr A = 4, det A = 4, so p(λ) = (λ − 2)². A − 2I = [[1, 1], [−1, −1]], which has the rank 1, so the geometric multiplicity is 1 and the eigenvector is v₁ = (1, −1). A generalized eigenvector w solves (A − 2I)w = v₁: w₁ + w₂ = 1, for example w = (1, 0). With P = [v₁ | w] = [[1, 1], [−1, 0]] (det P = 1, P⁻¹ = [[0, −1], [1, 1]]): AP = [[2, 3], [−2, −1]] and P⁻¹AP = [[0, −1], [1, 1]] · [[2, 3], [−2, −1]] = [[2, 1], [0, 2]] ✓.

Example 4: a 3 × 3 matrix with two eigenvalues

B = [[2, 1, 0], [0, 2, 0], [0, 0, 3]] is already in Jordan form: a block J₂(2) and a block J₁(3). The characteristic polynomial is (λ − 2)²(λ − 3) and the minimal polynomial is the same, since the largest block for λ = 2 has size 2.

How do you compute functions of a Jordan block?

Write J = λI + N with N nilpotent. Since λI commutes with N, the binomial theorem gives Jⁿ = λⁿI + nλⁿ⁻¹N + (n choose 2)λⁿ⁻²N² + …, a finite sum. For a 2 × 2 block, Jⁿ = [[λⁿ, nλⁿ⁻¹], [0, λⁿ]]. The exponential is e^tJ = e^λt [[1, t], [0, 1]]. Then Aⁿ = PJⁿP⁻¹ and e^tA = Pe^tJP⁻¹, which solve recurrences and the differential system x′ = Ax.

Example 5: a power of a Jordan block

[[2, 1], [0, 2]]³ = [[8, 3 · 4], [0, 8]] = [[8, 12], [0, 8]]. Direct check: [[2, 1], [0, 2]]² = [[4, 4], [0, 4]], and multiplying by [[2, 1], [0, 2]] gives [[8, 4 + 8], [0, 8]] = [[8, 12], [0, 8]] ✓.

Example 6: a differential system

For x′ = Ax with A of Example 3, the solutions combine e^2tv₁ and e^2t(t v₁ + w): the term t e^2t appears because of the Jordan block, as for a repeated root of a characteristic equation of a differential equation.

Table of Jordan types for a 4 × 4 matrix

The forms (3, 1) and (2, 2) differ in their kernel dimensions: for (3, 1), d₂ = 3; for (2, 2), d₂ = 4. The number of forms equals the number of partitions of 4, which is 5.

Where does the Jordan form come from?

Camille Jordan published the canonical form in his 1870 Traité des substitutions et des équations algébriques. Karl Weierstrass had classified matrices by elementary divisors in 1868, which is equivalent in content, and Frobenius later connected the approaches through the minimal polynomial. Over the real numbers a variant uses 2 × 2 blocks for complex conjugate pairs of eigenvalues.

Which mistakes are common?

Putting the ones below the diagonal

The convention used here places the ones on the superdiagonal. Some books use the subdiagonal, which corresponds to reversing the order of the chain.

Choosing a generalized eigenvector in the wrong kernel

The top vector of a chain must lie in ker (A − λI)^k but outside ker (A − λI)^k−1; otherwise the chain collapses.

Confusing algebraic and geometric multiplicities

The number of blocks is the geometric multiplicity, and the sum of the block sizes is the algebraic multiplicity.

Expecting a real Jordan form for complex eigenvalues

With non-real eigenvalues, the Jordan form exists over ℂ. Over ℝ, one uses the real Jordan form with rotation-like 2 × 2 blocks.