Chapter 26 Exercises — Complex Eigenvalues

Work the tiers in order; each builds on the last. ⭐ checks understanding, ⭐⭐ asks for hand computation, ⭐⭐⭐ asks for a proof (A) or code (C), and ⭐⭐⭐⭐ is an open-ended application. Problems marked [code] want a short numpy snippet; [proof] wants a rigorous argument in the §10 four-part style. Throughout, $A$ denotes a real matrix unless stated, $\lambda = re^{i\theta} = a + bi$ a complex eigenvalue, and $\mathbf{v} = \mathbf{p} + i\mathbf{q}$ its complex eigenvector with $\mathbf{p} = \operatorname{Re}(\mathbf{v})$, $\mathbf{q} = \operatorname{Im}(\mathbf{v})$.

Tier 1 — ⭐ Conceptual

26.1. In one sentence, explain geometrically why a rotation of the plane (by an angle that is not a multiple of $180°$) has no real eigenvectors.

26.2. A real $2\times 2$ matrix has eigenvalue $3 - 5i$. Write its other eigenvalue, and state the theorem that forces it.

26.3. Translate each into plain English: (a) $|\lambda| = 1$; (b) $|\lambda| > 1$; (c) $|\lambda| < 1$; (d) $\arg(\lambda) = 90°$. Each answer should mention rotation and/or scaling.

26.4. True or false, with a one-line reason: a real $3\times 3$ matrix can have eigenvalues $i$, $-i$, and $2i$.

26.5. Why can a single complex eigenvector not be drawn as one arrow in $\mathbb{R}^2$? What two real objects does it package instead?

26.6. Fill in the blanks: for a complex eigenvalue $\lambda = a + bi$ of a real matrix, the argument of $\lambda$ is the __ of the transformation and the modulus of $\lambda$ is the ____.

26.7. A real $2\times 2$ matrix has complex eigenvalues. What is the sign of $\operatorname{tr}(A)^2 - 4\det(A)$, and why?

26.8. Explain why the determinant of a real $2\times 2$ matrix with complex eigenvalues $re^{\pm i\theta}$ equals $r^2$, and why this is automatically a positive real number.

Tier 2 — ⭐⭐ Hand computation

26.9. For the rotation by $\theta = 60°$, write the characteristic polynomial $\lambda^2 - 2\cos\theta\,\lambda + 1$ explicitly, solve it, and confirm the eigenvalues are $e^{\pm i\,60°} = \tfrac12 \pm \tfrac{\sqrt3}{2}i$.

26.10. Find the eigenvalues of $A = \begin{psmallmatrix}3 & -1\\ 1 & \phantom{-}3\end{psmallmatrix}$ by hand (use trace and determinant). Then write the modulus $r$ and the argument $\theta$ in degrees.

26.11. For the same $A = \begin{psmallmatrix}3 & -1\\ 1 & \phantom{-}3\end{psmallmatrix}$, write the real canonical block $C = \begin{psmallmatrix}a & -b\\ b & a\end{psmallmatrix}$ directly from the eigenvalue $\lambda = a + bi$. (No eigenvector needed — why not?)

26.12. Decide, using only $\operatorname{tr}(A)^2 - 4\det(A)$, which of these have complex eigenvalues: (a) $\begin{psmallmatrix}2 & -5\\ 1 & 2\end{psmallmatrix}$, (b) $\begin{psmallmatrix}4 & 1\\ 1 & 2\end{psmallmatrix}$, (c) $\begin{psmallmatrix}0 & -1\\ 4 & 0\end{psmallmatrix}$, (d) $\begin{psmallmatrix}1 & 2\\ 0 & 3\end{psmallmatrix}$.

26.13. The matrix $\begin{psmallmatrix}0 & -3\\ 3 & 0\end{psmallmatrix}$ is already in block form. State its eigenvalues, modulus, and rotation angle by inspection, and describe in words what it does to the plane.

26.14. Compute, by hand, the complex eigenvector of the rotation by $90°$, $R = \begin{psmallmatrix}0 & -1\\ 1 & 0\end{psmallmatrix}$, for the eigenvalue $\lambda = i$. (Solve $(R - iI)\mathbf{v} = \mathbf{0}$.) Then write its real and imaginary parts.

26.15. For $\lambda = 1 + i$, compute $\lambda^2$, $\lambda^3$, and $\lambda^4$ in polar form $r^n e^{in\theta}$. What is the modulus after 4 steps, and how many total degrees has it rotated?

26.16. A real matrix has eigenvalues $0.8\,e^{\pm i\,30°}$. After how many applications does a typical vector's length first drop below half its original value? (Use $0.8^n < 0.5$.)

Tier 3 — ⭐⭐⭐ Proof (A) / Coding (C)

26.17. [proof] Prove that if $A$ is real and $\lambda$ is a real eigenvalue, then $A$ has a real eigenvector for $\lambda$. (Contrast with the complex case; use that $A - \lambda I$ is then a real singular matrix.)

26.18. [proof] Prove that the eigenvalues of a real orthogonal matrix all have modulus $1$. (Hint: an orthogonal $Q$ preserves length; apply $\lVert Q\mathbf{v}\rVert = \lVert\mathbf{v}\rVert$ to a complex eigenvector, using the complex norm $\lVert\mathbf{v}\rVert^2 = \mathbf{v}^{*}\mathbf{v}$ and the fact that $Q^{*} = Q^{\mathsf{T}} = Q^{-1}$ for a real orthogonal matrix.)

26.19. [proof] Starting from $A\mathbf{v} = \lambda\mathbf{v}$ with $\mathbf{v} = \mathbf{p} + i\mathbf{q}$ and $\lambda = a + bi$, derive the two real equations $A\mathbf{p} = a\mathbf{p} - b\mathbf{q}$ and $A\mathbf{q} = b\mathbf{p} + a\mathbf{q}$, and explain why they say $AP = PC$ for $P = [\mathbf{p}\mid-\mathbf{q}]$ and $C = \begin{psmallmatrix}a & -b\\ b & a\end{psmallmatrix}$.

26.20. [proof] Prove that a real matrix of odd dimension $n$ has at least one real eigenvalue. (Use the conjugate-pair theorem and a counting argument on the $n$ eigenvalues.)

26.21. [code] Write a numpy snippet that takes any real $2\times 2$ matrix, computes its eigenvalues with np.linalg.eig, and prints whether the matrix "rotates" (complex eigenvalues), giving the modulus and the rotation angle in degrees when it does. Test it on $\begin{psmallmatrix}1 & -2\\ 1 & 3\end{psmallmatrix}$ and $\begin{psmallmatrix}3 & 0\\ 0 & 2\end{psmallmatrix}$.

26.22. [code] Implement rotation_scaling(A) exactly as described in the chapter's Build-Your-Toolkit callout (pure Python, math only). Verify on three matrices that your $r$, $\theta$, and reconstructed block agree with np.linalg.eig and that $P C P^{-1} = A$ to tolerance.

26.23. [code] Write a snippet that builds the matrix "scale by $r$, rotate by $\theta$" for user-chosen $r$ and $\theta$, iterates it 20 times from $(1, 0)$, and plots the trajectory. Run it for $(r, \theta) = (1.1, 20°)$, $(1.0, 20°)$, and $(0.9, 20°)$, and describe the three spirals.

Tier 4 — ⭐⭐⭐⭐ Application

26.24. [code] Damped oscillator. A discrete model of a mass on a spring with friction has state matrix $A = \begin{psmallmatrix}0.95 & -0.3\\ 0.3 & \phantom{-}0.95\end{psmallmatrix}$. (a) Find its eigenvalues, modulus, and rotation angle. (b) Is the system stable? (c) Estimate the number of steps per full oscillation (one $360°$ turn) and the per-step decay factor. (d) Plot $\lVert\mathbf{x}_n\rVert$ versus $n$ and confirm it decays like $r^n$.

26.25. [code] Predator–prey cycle. A linearized predator–prey model near equilibrium is $A = \begin{psmallmatrix}1.0 & -0.5\\ 0.4 & \phantom{-}1.0\end{psmallmatrix}$. (a) Show the eigenvalues are complex and find $r$ and $\theta$. (b) Is the cycle growing, decaying, or steady? (c) Interpret the argument $\theta$ as the cycle's angular speed: roughly how many steps does one boom-and-bust cycle take? (d) Discuss what management action would push $r$ toward $1$ (a sustained cycle) or below $1$ (a damped approach to equilibrium).

26.26. Why complex, why now. In two or three paragraphs, explain to a classmate who has finished Chapter 25 (diagonalization) but not this chapter: (a) why diagonalization "fails" over the reals for a rotation but succeeds over $\mathbb{C}$; (b) what the real canonical block form buys you in exchange for giving up a diagonal matrix; and (c) how the modulus of a complex eigenvalue connects to the long-run behavior of matrix powers $A^n$. Tie your answer to the recurring theme that eigenvalues reveal what a matrix really does.

Hints and selected answers

26.2

The other eigenvalue is $3 + 5i$, the complex conjugate, by the conjugate-pair theorem for real matrices (a complex eigenvalue of a real matrix always comes with its conjugate).

26.4

False. Complex eigenvalues of a real matrix come in conjugate pairs; $i$ pairs with $-i$, but $2i$ would need its conjugate $-2i$ as a fourth eigenvalue — impossible for a $3\times 3$ matrix (only three eigenvalues), and three eigenvalues cannot all be non-real, so at least one must be real.

26.10

$\operatorname{tr}(A) = 6$, $\det(A) = 9 + 1 = 10$, discriminant $36 - 40 = -4 < 0$. Eigenvalues $\lambda = 3 \pm i$. Modulus $r = \sqrt{10} \approx 3.162$; argument $\theta = \operatorname{atan2}(1, 3) \approx 18.43°$.

26.11

$C = \begin{psmallmatrix}3 & -1\\ 1 & \phantom{-}3\end{psmallmatrix}$ — and notice $A$ already equals its own block form, because it already has equal diagonal entries and equal-but-opposite off-diagonals. No eigenvector is needed because the block is read straight off $\lambda = a + bi = 3 + i$ ($a = 3$, $b = 1$).

26.12

Compute $\operatorname{tr}^2 - 4\det$: (a) $4^2 - 4(4 + 5) = 16 - 36 = -20 < 0$ → complex; (b) $6^2 - 4(8 - 1) = 36 - 28 = 8 > 0$ → real; (c) $0^2 - 4(0 + 4) = -16 < 0$ → complex; (d) $4^2 - 4(3) = 4 > 0$ → real. So (a) and (c) rotate.

26.13

Block with $a = 0$, $b = 3$: eigenvalues $\pm 3i$, modulus $r = 3$, argument $\theta = 90°$. It rotates the plane a quarter-turn counterclockwise and scales it by $3$.

26.14

$(R - iI) = \begin{psmallmatrix}-i & -1\\ 1 & -i\end{psmallmatrix}$. Top row: $-i\,v_1 - v_2 = 0 \Rightarrow v_2 = -i\,v_1$. Take $\mathbf{v} = (1, -i)$. Real part $(1, 0)$, imaginary part $(0, -1)$. (Same eigenvector as the $30°$ rotation in the chapter — every $2\times 2$ rotation shares the eigenvector $(1, \mp i)$.)

26.16

$0.8^n < 0.5 \Rightarrow n > \ln(0.5)/\ln(0.8) \approx 3.106$, so at $n = 4$ the length first drops below half (the rotation does not affect length; only the modulus $0.8$ matters).

26.18 (sketch)

Let $Q\mathbf{v} = \lambda\mathbf{v}$ with $\mathbf{v} \ne \mathbf{0}$ complex. Then $\lVert Q\mathbf{v}\rVert^2 = (Q\mathbf{v})^{*}(Q\mathbf{v}) = \mathbf{v}^{*}Q^{*}Q\mathbf{v} = \mathbf{v}^{*}\mathbf{v} = \lVert\mathbf{v}\rVert^2$ using $Q^{*}Q = Q^{\mathsf{T}}Q = I$. But also $\lVert Q\mathbf{v}\rVert^2 = \lVert\lambda\mathbf{v}\rVert^2 = |\lambda|^2\lVert\mathbf{v}\rVert^2$. Cancel $\lVert\mathbf{v}\rVert^2 \ne 0$ to get $|\lambda|^2 = 1$, so $|\lambda| = 1$.

26.24

(a) $\operatorname{tr} = 1.9$, $\det = 0.95^2 + 0.09 = 0.9925$; eigenvalues $0.95 \pm 0.3i$, $r = \sqrt{0.9925} \approx 0.9962$, $\theta = \operatorname{atan2}(0.3, 0.95) \approx 17.5°$. (b) Stable: $r < 1$, oscillations decay. (c) About $360/17.5 \approx 20.6$ steps per oscillation; per-step decay factor $\approx 0.9962$ (very slow decay — lightly damped). (d) The plotted $\lVert\mathbf{x}_n\rVert$ should track $r^n$ closely.

26.25

(a) $\operatorname{tr} = 2.0$, $\det = 1.0 + 0.2 = 1.2$, discriminant $4 - 4.8 = -0.8 < 0$ → complex; eigenvalues $1 \pm \sqrt{0.2}\,i \approx 1 \pm 0.447i$, $r = \sqrt{1.2} \approx 1.095$, $\theta \approx 24.1°$. (b) Growing: $r > 1$, so oscillations amplify away from equilibrium. (c) About $360/24.1 \approx 14.9$ steps per cycle. (d) Reducing the off-diagonal coupling or adding a self-limiting (negative-diagonal) term lowers $\det$ relative to $\operatorname{tr}$, pulling $r$ toward or below $1$.