Chapter 36 Exercises — Jordan Normal Form

Work these with pencil first, then confirm the computational ones with numpy or sympy. Tiers: ⭐ conceptual · ⭐⭐ computation (hand) · ⭐⭐⭐ proof (A) / coding (C) · ⭐⭐⭐⭐ application/synthesis. Throughout, $A$ is a square matrix, $\lambda$ an eigenvalue, $N = A - \lambda I$, and $J_k(\lambda)$ the $k\times k$ Jordan block. Unless stated, you may work over $\mathbb{C}$.

⭐ Conceptual

36.1. In one sentence each, define algebraic multiplicity and geometric multiplicity of an eigenvalue, and state the inequality that always relates them. Which one being strictly smaller makes a matrix defective?

36.2. True or false, with a one-sentence reason each: (a) Every $2\times 2$ matrix with a repeated eigenvalue is defective. (b) A defective matrix must be singular. (c) A matrix with $n$ distinct eigenvalues can never be defective. (d) A real symmetric matrix can be defective. (e) The Jordan form of a diagonalizable matrix is diagonal.

36.3. Explain in two or three sentences why a Jordan block $\begin{bmatrix}\lambda&1\\0&\lambda\end{bmatrix}$ is described as a "shear in the eigen-direction." What does the off-diagonal $1$ represent geometrically, and why can no eigenvector capture it?

36.4. A generalized eigenvector $\mathbf{w}$ is found by solving which equation? How does that equation differ from the one an ordinary eigenvector solves, and what condition on $\mathbf{v}$ must hold for the generalized equation to be solvable?

36.5. State the Jordan canonical form theorem, including its conditions. Over which number system is it guaranteed to exist, and why does the guarantee fail over $\mathbb{R}$? In what sense is the form "unique"?

36.6. For an eigenvalue $\lambda$, what does the geometric multiplicity tell you about the number of Jordan blocks for $\lambda$, and what does the algebraic multiplicity tell you about their total size? Why do these two numbers not always determine the individual block sizes?

36.7. Why is the Jordan form almost never computed by numerical software, even though it exists for every matrix? Name the property of defectiveness that makes it numerically fragile, and name the stable decomposition that production code uses in its place.

36.8. Explain why diagonalization is "the special case of the Jordan form where every block has size $1$." If you are told a matrix's Jordan form has a block of size $\ge 2$, what can you immediately conclude about diagonalizability?

⭐⭐ Computation (by hand, then check with numpy/sympy)

36.9. For $A = \begin{bmatrix}3&1\\0&3\end{bmatrix}$: find the eigenvalue and its algebraic multiplicity, compute $\dim N(A-3I)$ to get the geometric multiplicity, and state whether $A$ is defective. (You should find it is — and that it is already in Jordan form, so $P = I$.)

36.10. For $A = \begin{bmatrix}5&0\\0&5\end{bmatrix}$: repeat the diagnosis of 36.9. Is this matrix defective? Why does a repeated eigenvalue not force defectiveness here? What is its Jordan form?

36.11. Diagnose $A = \begin{bmatrix}4&1\\-1&2\end{bmatrix}$. Show the characteristic polynomial is $(\lambda-3)^2$, find the single eigenvector, solve $(A-3I)\mathbf{w}=\mathbf{v}$ for a generalized eigenvector, assemble $P$, and verify $P^{-1}AP = \begin{bmatrix}3&1\\0&3\end{bmatrix}$ by checking $AP = PJ$ (no inversion needed). Confirm with sympy.Matrix([[4,1],[-1,2]]).jordan_form().

36.12. Repeat the full procedure of 36.11 for $A = \begin{bmatrix}0&1\\-1&2\end{bmatrix}$ (a companion matrix; you should find $\lambda=1$ doubled, defective). Find the chain and the Jordan form.

36.13. For the $3\times 3$ block $A = \begin{bmatrix}2&1&0\\0&2&1\\0&0&2\end{bmatrix}$: confirm $\lambda=2$ with algebraic multiplicity $3$, compute $\dim N(A-2I)$, $\operatorname{rank}(A-2I)$, $\operatorname{rank}((A-2I)^2)$, and $(A-2I)^3$. Use the rank formula of §36.6 to confirm there is a single block of size $3$.

36.14. Compute $\begin{bmatrix}3&1\\0&3\end{bmatrix}^4$ by hand using the formula $J^m = \begin{bmatrix}\lambda^m & m\lambda^{m-1}\\0&\lambda^m\end{bmatrix}$, then verify with np.linalg.matrix_power. What are the diagonal and off-diagonal entries?

36.15. Using $J_2(\lambda) = \lambda I + N$ with $N^2=0$, derive $e^{J t}$ for $J = \begin{bmatrix}-1&1\\0&-1\end{bmatrix}$ and $t=1$. Give the entries as numbers (in terms of $e^{-1}$), then check against scipy.linalg.expm.

36.16. A matrix has eigenvalue $\lambda=4$ with algebraic multiplicity $5$ and geometric multiplicity $2$. List all the possible Jordan block-size patterns for $\lambda=4$ consistent with these two numbers. (Hint: two blocks, total size $5$.) What additional information would pin down which pattern is correct?

36.17. For $A = \begin{bmatrix}2&1&0\\0&2&0\\0&1&2\end{bmatrix}$ (the §36.4 example): verify that the eigenvector $\mathbf{e}_1=(1,0,0)$ does not lie in the column space of $A-2I$, so $(A-2I)\mathbf{w}=\mathbf{e}_1$ has no solution. Then verify that $\mathbf{v}=(1,0,1)$ does lie in the column space, and is the correct chain-starting eigenvector.

⭐⭐⭐ Proof (A) and Coding (C)

36.18. (Proof) Prove that in a Jordan chain $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ (with $(A-\lambda I)\mathbf{v}_1=\mathbf{0}$ and $(A-\lambda I)\mathbf{v}_{j}=\mathbf{v}_{j-1}$ for $j\ge 2$), the vectors are linearly independent. (Hint: suppose $\sum c_j\mathbf{v}_j = \mathbf{0}$ and apply $(A-\lambda I)^{k-1}$ to show $c_k=0$, then $(A-\lambda I)^{k-2}$, and so on.)

36.19. (Proof) Prove the multiplicity inequality used throughout: for any eigenvalue $\lambda$, the geometric multiplicity $\dim N(A-\lambda I)$ is at least $1$ and at most the algebraic multiplicity. (You proved the upper bound in Chapter 24; restate the argument, e.g. via the fact that a basis of the eigenspace extends to a basis in which $A$ is block-triangular.)

36.20. (Proof) Show that a Jordan block $J_k(\lambda)$ satisfies $(J_k(\lambda) - \lambda I)^k = 0$ but $(J_k(\lambda) - \lambda I)^{k-1} \neq 0$. Conclude that the size of the largest Jordan block for $\lambda$ equals the index of $\lambda$ — the smallest $p$ with $\operatorname{rank}((A-\lambda I)^p) = \operatorname{rank}((A-\lambda I)^{p+1})$.

36.21. (Proof) Prove that two diagonalizable matrices are similar if and only if they have the same eigenvalues with the same algebraic multiplicities. Then exhibit two defective matrices with identical eigenvalues and identical algebraic and geometric multiplicities that are nonetheless not similar. (Hint: you need at least a $4\times 4$ example where block sizes $3+1$ versus $2+2$ differ; explain why eigenvalue data alone cannot tell them apart but the Jordan form can.)

36.22. (Proof) Using the binomial expansion of $(\lambda I + N)^m$ with $N^k = 0$, prove that the $(i,j)$ entry of $J_k(\lambda)^m$ (for $j \ge i$) is $\binom{m}{j-i}\lambda^{m-(j-i)}$, and zero for $j < i$. Conclude that every entry of $J_k(\lambda)^m$ is a polynomial in $m$ of degree at most $k-1$ times $\lambda^m$.

36.23. (Coding) Implement is_defective(A, tol=1e-9) in pure Python that returns True when some eigenvalue's geometric multiplicity is strictly less than its algebraic multiplicity. Use numpy only to get eigenvalues and ranks (computing geometric multiplicity as $n - \operatorname{rank}(A-\lambda I)$, grouping near-equal eigenvalues within tol). Test it returns True for $\begin{bmatrix}2&1\\0&2\end{bmatrix}$ and False for $\begin{bmatrix}5&0\\0&5\end{bmatrix}$ and any matrix with distinct eigenvalues.

36.24. (Coding) This is the chapter's Build Your Toolkit task. Implement jordan_chain(A, lam, tol=1e-9) in toolkit/jordan.py (pure Python — reuse your toolkit/linear_systems.py solver from Chapter 4): find an eigenvector, solve $(A-\lambda I)\mathbf{w}=\mathbf{v}$ for the generalized eigenvector, assemble $P$, and return P, J with J = P^{-1}AP. Verify on $\begin{bmatrix}1&-1\\1&3\end{bmatrix}$ and the $3\times3$ example of §36.4 that your J matches the hand result and sympy's jordan_form() (the $J$'s agree; your $P$ may differ — explain why). Add a guard that warns when the matrix is diagonalizable (no chain needed).

36.25. (Coding) Demonstrate numerical fragility empirically. Start from $A = \begin{bmatrix}2&1\\0&2\end{bmatrix}$ and add perturbations $\epsilon \in \{10^{-4}, 10^{-8}, 10^{-12}\}$ to the bottom-left entry. For each, compute the two eigenvalues and the gap between them, and the condition number of the eigenvector matrix. Confirm numerically that the eigenvalue gap scales like $\sqrt{\epsilon}$ (not $\epsilon$) and that the condition number blows up as $\epsilon \to 0$. Write one sentence explaining why this means a floating-point routine cannot reliably detect defectiveness.

⭐⭐⭐⭐ Application / Synthesis

36.26. (Engineering — critical damping) A damped oscillator obeys $x'' + 2c\,x' + k\,x = 0$, with companion matrix $A = \begin{bmatrix}0&1\\-k&-2c\end{bmatrix}$ for the state $(x, x')$. (a) Show the eigenvalues are repeated exactly when $c^2 = k$ (critical damping). (b) For $c=3, k=9$, write $A$, confirm the double eigenvalue $-3$ is defective, and find the Jordan chain. (c) Using $e^{Jt} = e^{-3t}\begin{bmatrix}1&t\\0&1\end{bmatrix}$, explain why the displacement contains a $t\,e^{-3t}$ term, and why that term is the mathematical reason a critically-damped door-closer "creeps" to zero rather than oscillating or crawling. Tie this to Chapter 37's matrix-exponential solution.

36.27. (Data / transition models) A two-stage cascade moves tokens from stage A to stage B, each stage retaining a fraction $r$ per step and stage A feeding stage B, giving transition matrix $T = \begin{bmatrix}r&0\\r&r\end{bmatrix}$. (a) Show $T$ is defective with double eigenvalue $r$ and a single eigenvector. (b) Find the Jordan form. (c) Using $T^m$, show the off-diagonal (A-to-B accumulation) entry grows like $m\,r^{m-1}$ — a polynomial-times-geometric term. (d) Contrast with a diagonalizable transition matrix (distinct retention rates), whose entries decay as pure $\lambda^m$, and explain in words what the extra factor of $m$ says about how tokens pile up in stage B before the decay wins.

36.28. (Synthesis) A colleague says: "Defective matrices are just numerical noise — perturb any matrix slightly and it becomes diagonalizable, so the Jordan form is useless." Write a paragraph rebutting this. Address: (a) why some defective matrices are exact by design (critical damping, repeated natural frequencies), (b) why perturbing changes the physics rather than fixing it, (c) why the Jordan form is essential for understanding powers and exponentials even if never computed numerically, and (d) the legitimate kernel of truth in the colleague's claim (numerical fragility, §36.8). Reference at least three specific results from this chapter.