Questions: Jordan Normal Form and Generalized Eigenvectors

Question 1

A student computes that matrix A has eigenvalue λ = 5 with algebraic multiplicity 3. She concludes that A must have a single 3×3 Jordan block for λ = 5. What is wrong with this reasoning?

• A. Algebraic multiplicity cannot exceed 2 in a real matrix
• B. The sizes of Jordan blocks depend on the geometric multiplicity (dimension of the eigenspace), not just the algebraic multiplicity. If the eigenspace is three-dimensional, A is diagonalizable and all three blocks are 1×1. A single 3×3 block only arises when the eigenspace is one-dimensional.
• C. Jordan blocks can only appear for complex eigenvalues, not real ones
• D. Algebraic multiplicity 3 guarantees exactly three separate 1×1 Jordan blocks

Answer: (B)

Algebraic multiplicity tells you the total 'size budget' for Jordan blocks of eigenvalue λ — their sizes must sum to it. The geometric multiplicity tells you how many blocks there are. One 3×3 block (geometric multiplicity 1), three 1×1 blocks (geometric multiplicity 3), or a 2×2 and a 1×1 block (geometric multiplicity 2) are all consistent with algebraic multiplicity 3. The student has conflated the two multiplicities.

Question 2

What distinguishes a generalized eigenvector vᵢ from a true eigenvector v₁ in a Jordan chain for eigenvalue λ?

• A. A generalized eigenvector satisfies Avᵢ = λvᵢ, just like a true eigenvector
• B. A generalized eigenvector satisfies (A − λI)vᵢ = vᵢ₋₁ — applying (A − λI) to it yields the previous vector in the chain rather than zero
• C. A generalized eigenvector must have unit length
• D. A generalized eigenvector spans the same subspace as the true eigenvector for λ

Answer: (B)

A true eigenvector satisfies (A − λI)v₁ = 0 — it is annihilated by (A − λI). A generalized eigenvector at level i satisfies (A − λI)vᵢ = vᵢ₋₁: applying the defect operator maps it to the previous vector in the chain, not to zero. This chain structure is precisely what Jordan blocks encode — the 1s on the superdiagonal represent these 'shift' relationships between consecutive chain vectors.

Question 3

A matrix is diagonalizable if and mainly if most of its eigenvalues are distinct (no repeated eigenvalues).

• T. True
• F. False

Answer: False

Distinct eigenvalues are sufficient but not necessary for diagonalizability. A matrix with repeated eigenvalues can still be diagonalizable if the geometric multiplicity equals the algebraic multiplicity for every eigenvalue — that is, if there are enough linearly independent eigenvectors to form a full basis. For example, the identity matrix has only one eigenvalue (λ = 1 with algebraic multiplicity n) but is trivially diagonalizable. The key test is the multiplicity comparison, not whether eigenvalues are distinct.

Question 4

The number of distinct Jordan blocks for an eigenvalue λ in the Jordan normal form equals the geometric multiplicity of λ.

• T. True
• F. False

Answer: True

Each Jordan block begins with a true eigenvector — you need one eigenvector to start each chain. Since the eigenvectors for λ are exactly the nonzero vectors in ker(A − λI), and the number of linearly independent eigenvectors is the geometric multiplicity, that equals the number of Jordan blocks. Their sizes sum to the algebraic multiplicity. This relationship between the two multiplicities completely determines the Jordan block structure.

Question 5

Why does a defective matrix (one that cannot be diagonalized) produce solutions to differential equations y' = Ay that include polynomial terms like te^(λt), rather than pure exponentials?

Model answer: When A has a Jordan block of size k for eigenvalue λ, computing the matrix exponential e^(At) requires raising the Jordan block to a power. The 1s on the superdiagonal, combined with the binomial theorem, generate polynomial terms: a 2×2 block produces t·e^(λt), a 3×3 block produces t·e^(λt) and t²·e^(λt), and so on. The polynomial factors arise because there are not enough true eigenvectors — the generalized eigenvectors fill the solution space, but they interact with the defect structure of A to produce polynomial-exponential solutions rather than pure exponentials.

This connects Jordan form to differential equations and explains why defective systems matter in applications. The polynomial growth from Jordan blocks is not a curiosity — it appears in critical phenomena like resonance in mechanics and repeated-pole behavior in control systems. Recognizing the Jordan block as the algebraic source of polynomial-exponential solutions is the payoff of the entire theory.