Exercises — Chapter 23: Eigenvalues and Eigenvectors

Work these with pencil first; reach for numpy only where a problem says (code) or to check a hand result. Eigenvectors are determined only up to a nonzero scalar, so your representative may differ from the key by a scale factor or a sign — that is correct. Difficulty tiers: ⭐ conceptual · ⭐⭐ hand computation · ⭐⭐⭐ proof (A) / coding (C) · ⭐⭐⭐⭐ application.

⭐ Tier 1 — Conceptual (the picture before the algebra)

23.1 In one sentence, state what it means geometrically for a nonzero vector $\mathbf{v}$ to be an eigenvector of a matrix $A$. Your answer should mention the word direction.

23.2 True or false, with a one-line reason each: (a) The zero vector is an eigenvector of every matrix. (b) $0$ can be an eigenvalue of a matrix. (c) If $\mathbf{v}$ is an eigenvector, so is $-\mathbf{v}$. (d) Every $2\times2$ real matrix has two real eigenvectors.

23.3 A matrix $A$ has an eigenvector $\mathbf{v}$ with eigenvalue $\lambda = -2$. Describe in words what $A$ does to $\mathbf{v}$: does it grow, shrink, flip, or fix the vector? Does $\mathbf{v}$ stay on its own line?

23.4 Match each eigenvalue to the behavior of its eigen-direction under repeated application of $A$: (i) $\lambda = 1$, (ii) $\lambda = 0.5$, (iii) $\lambda = 3$, (iv) $\lambda = 0$, (v) $\lambda = -1$. Behaviors: grows without bound; shrinks toward the origin; stays fixed; collapses to the origin in one step; flips back and forth without changing length.

23.5 Explain why the eigenspace $E_\lambda$ is allowed to contain the zero vector even though the zero vector is not an eigenvector. (One or two sentences; mention the word subspace.)

23.6 A classmate says, "I computed an eigenvalue of $-4$, so I must have made a sign error — eigenvalues are stretch factors and stretch factors are positive." Correct them.

23.7 Without any computation, give the eigenvalues of (a) the $3\times3$ identity $I_3$, and (b) the scaling matrix $7I_3$. What is the eigenspace in each case?

⭐⭐ Tier 2 — Hand computation

23.8 For $A = \begin{psmallmatrix}2&1\\1&2\end{psmallmatrix}$ (the chapter's home matrix), verify directly by matrix–vector multiplication that: (a) $(3,3)$ is an eigenvector and state its eigenvalue; (b) $(-2,2)$ is an eigenvector and state its eigenvalue; (c) $(1,0)$ is not an eigenvector.

23.9 Find the eigenvalues of $A = \begin{psmallmatrix}3&0\\0&-5\end{psmallmatrix}$ by inspection (it is diagonal), and give an eigenvector for each.

23.10 Use the trace–determinant shortcut to find the eigenvalues of each $2\times2$ matrix (find two numbers that sum to the trace and multiply to the determinant), then state them: (a) $\begin{psmallmatrix}5&2\\2&5\end{psmallmatrix}$ (b) $\begin{psmallmatrix}1&6\\1&2\end{psmallmatrix}$ (c) $\begin{psmallmatrix}4&0\\3&4\end{psmallmatrix}$.

23.11 For $A = \begin{psmallmatrix}4&1\\2&3\end{psmallmatrix}$ (worked in §23.6), recompute the eigenvalues from $\det(A-\lambda I)=0$, then find an eigenvector for each by solving $(A-\lambda I)\mathbf{v}=\mathbf{0}$. Confirm $\lambda_1+\lambda_2=\operatorname{tr}(A)$ and $\lambda_1\lambda_2=\det(A)$.

23.12 Find all eigenvalues and a basis for each eigenspace of $A = \begin{psmallmatrix}1&2\\2&1\end{psmallmatrix}$. Describe the two eigen-lines geometrically (give their angles to the $x$-axis).

23.13 The reflection across the line $y=x$ has matrix $A = \begin{psmallmatrix}0&1\\1&0\end{psmallmatrix}$. Predict its eigenvalues from the geometry (what does a mirror fix? what does it flip?), then confirm by computing $\det(A-\lambda I)=0$ and find an eigenvector for each.

23.14 Show by direct computation that the projection $A = \begin{psmallmatrix}1&0\\0&0\end{psmallmatrix}$ has eigenvalues $1$ and $0$. Identify the eigenvector with eigenvalue $0$ and explain its connection to the null space $N(A)$.

23.15 The matrix $A = \begin{psmallmatrix}1&1\\0&1\end{psmallmatrix}$ (a shear) has characteristic equation $(1-\lambda)^2=0$. Find all its eigenvalues and all its eigenvectors. How many independent eigen-directions does it have? (This is the defective case from §23.5.2.)

23.16 Using the eigen-decomposition trick of §23.7: for the home matrix $A=\begin{psmallmatrix}2&1\\1&2\end{psmallmatrix}$ with eigenpairs $(3,(1,1))$ and $(1,(-1,1))$, write $(4,2)$ in the eigenbasis and use it to compute $A(4,2)$ and $A^4(4,2)$ without multiplying matrices.

⭐⭐⭐ Tier 3 — Proof (A) and coding (C)

23.17 (proof) Prove that if $\mathbf{v}$ is an eigenvector of $A$ with eigenvalue $\lambda$, then for any scalar $c\ne 0$, $c\mathbf{v}$ is also an eigenvector with the same eigenvalue $\lambda$. Where does linearity enter?

23.18 (proof) Prove that $\lambda = 0$ is an eigenvalue of a square matrix $A$ if and only if $A$ is singular (i.e., $\det(A)=0$). Hint: relate the eigenspace $E_0$ to the null space $N(A)$.

23.19 (proof) Suppose $\mathbf{v}$ is an eigenvector of $A$ with eigenvalue $\lambda$. Prove that $\mathbf{v}$ is also an eigenvector of $A^2$, and find its eigenvalue. Generalize: what is the eigenvalue of $\mathbf{v}$ under $A^k$? Under $A^{-1}$ (assuming $A$ invertible)?

23.20 (proof) Prove that eigenvectors belonging to distinct eigenvalues are linearly independent, for the case of two eigenvalues. That is, if $A\mathbf{v}_1=\lambda_1\mathbf{v}_1$ and $A\mathbf{v}_2=\lambda_2\mathbf{v}_2$ with $\lambda_1\ne\lambda_2$ and both eigenvectors nonzero, show $\{\mathbf{v}_1,\mathbf{v}_2\}$ is independent. Hint: assume $a\mathbf{v}_1+b\mathbf{v}_2=\mathbf{0}$, apply $A$, and also multiply the original by $\lambda_1$.

23.21 (proof) Show that $A$ and its transpose $A^{\mathsf{T}}$ have the same eigenvalues. Hint: $\det(A^{\mathsf{T}}-\lambda I) = \det((A-\lambda I)^{\mathsf{T}})$ and the determinant is invariant under transpose (Chapter 11). (Do they have the same eigenvectors? Comment.)

23.22 (code) Implement power_iteration(A, num_iters=1000, tol=1e-10) in toolkit/eigen.py in pure Python (use your Chapter 2 toolkit's vector operations, no numpy in the body). Return the dominant eigenvalue (via the Rayleigh quotient) and a unit dominant eigenvector. Verify on $\begin{psmallmatrix}4&1\\2&3\end{psmallmatrix}$ that you recover $\lambda\approx 5$ and a vector along $(1,1)$, and check against np.linalg.eig.

23.23 (code) Write a short numpy script that, given a $2\times2$ matrix, overlays its real eigen-lines on the recurring visualize_2d from Chapter 1 (import the function unchanged). Test it on $\begin{psmallmatrix}2&1\\1&2\end{psmallmatrix}$ and on $\begin{psmallmatrix}0&-1\\1&0\end{psmallmatrix}$; explain what you see (or fail to see) in each case.

23.24 (code) Verify the claim of §23.8.1 numerically: pick any invertible $P$, form $B = P^{-1}AP$ for $A=\begin{psmallmatrix}2&1\\1&2\end{psmallmatrix}$, and confirm with np.linalg.eig that $A$ and $B$ have identical eigenvalues but different entries. Repeat for three different $P$'s.

⭐⭐⭐⭐ Tier 4 — Application

23.25 (Markov steady state) A subscription service finds that each month $85\%$ of subscribers stay and $15\%$ cancel, while $30\%$ of the "cancelled" pool resubscribes and $70\%$ stays gone. With state $(\text{subscribed}, \text{cancelled})$, write the column-stochastic transition matrix $P$. Find the eigenvector for $\lambda=1$ and normalize it to sum to $1$. What fraction of the population is subscribed in the long run, regardless of the starting mix? Confirm $\lambda=1$ is an eigenvalue and that the other eigenvalue has $|\lambda|<1$.

23.26 (population growth / Leslie-style) A simplified two-stage insect population has juveniles $J$ and adults $A$ evolving by $\begin{psmallmatrix}J'\\A'\end{psmallmatrix} = \begin{psmallmatrix}1&4\\0.25&0\end{psmallmatrix}\begin{psmallmatrix}J\\A\end{psmallmatrix}$ each season (juveniles persist and each adult produces 4 new juveniles; a quarter of juveniles mature into adults). Show the characteristic equation is $\lambda^2-\lambda-1=0$ — so the eigenvalues are the golden ratio $\varphi\approx1.618$ and $-0.618$, exactly as in §23.6.2. What is the long-run growth rate $|\lambda_{\max}|$ per season — is the population growing or shrinking? Find the dominant eigenvector and interpret it as the stable juvenile-to-adult ratio (you should get $4\varphi:1\approx6.47:1$). (code) Confirm with power iteration that the population's growth factor approaches $\varphi$ each season.

23.27 (vibration / two-mass system) Two equal masses connected by identical springs lead to the stiffness matrix $K = \begin{psmallmatrix}2&-1\\-1&2\end{psmallmatrix}$. Find its eigenvalues and eigenvectors. The eigenvectors are the normal modes: describe the motion each one represents (do the masses move together or oppositely?), and explain why the eigenvalues correspond to the squared natural frequencies. (This anticipates case study 2.)

23.28 (PageRank teaser) Three web pages link as follows: page 1 links to pages 2 and 3; page 2 links to page 3; page 3 links to page 1. Build the column-stochastic link matrix (each page splits its "vote" equally among its out-links). (code) Run power iteration to find the dominant eigenvector (the PageRank scores) and rank the three pages. Which page is most important, and does the ranking match your intuition about which page receives the most "link weight"? Compare to np.linalg.eig.