Chapter 26 Quiz — Complex Eigenvalues

Q: A real matrix has eigenvalue . What is its other eigenvalue, and why must this be so? >

> The conjugate . The conjugate-pair theorem says complex eigenvalues of a real matrix always come in conjugate pairs (conjugating the eigen-equation leaves the real unchanged and produces ). Realness of the matrix is the load-bearing condition. >

Q: A real matrix has all its eigenvalues with modulus . What happens to a typical vector under repeated application ? >

> It spirals inward to the origin, shrinking by a factor and rotating by each step — a damped oscillation. The system is stable. Modulus below 1 means the spiral winds in. >

Q: What is the real canonical (block) form of a real matrix with eigenvalues , and what transformation does that block perform? >

> , which equals with , — a rotation by combined with a scaling by . The block is rotation-scaling in plain real arithmetic. >

Q: For a real matrix, how can you tell from the trace and determinant alone whether it has complex eigenvalues? >

> Compute the discriminant . If it is negative, the eigenvalues are a complex conjugate pair (the matrix rotates); if positive, two real eigenvalues; if zero, a repeated real eigenvalue. Negative discriminant = rotation. >

Q: Why must a real matrix have at least one real eigenvalue? >

> It has three eigenvalues, and complex ones come in conjugate pairs. Three values cannot be partitioned entirely into pairs (an odd count), so at least one must be its own conjugate — i.e., real. Odd dimension forces a leftover real eigenvalue (the fixed axis of a 3D rotation). >

Q: All eigenvalues of a real orthogonal matrix (a rotation/reflection) have what modulus, and why? >

> Modulus exactly : an orthogonal matrix preserves length, so it cannot scale any direction — the only allowed scale factor is . Hence every eigenvalue lies on the unit circle (, or the real values ). Length preservation pins the eigenvalues to the unit circle. >

Q: For a real matrix with complex eigenvalues , what is in terms of ? Explain geometrically. >

> . The determinant is the product of the eigenvalues, ; geometrically, the transformation scales every length by , so it scales 2D area by , and the determinant is exactly the area-scaling factor (Chapter 11). Determinant is the squared modulus because area scales as length squared. >

Q: When you compute the powers of a real matrix with complex eigenvalues via the block form, what happens to the modulus and the angle? >

> The modulus is raised to the -th power and the angle is multiplied by : , mirroring (De Moivre). steps means scale by , rotate by . >

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Chapter 26 Quiz — Complex Eigenvalues

Twelve conceptual checks. Try each before opening the answer. One-line explanations follow each solution.

Q1. Why does a rotation of the plane by $30°$ have no real eigenvectors?

Answer
Because a rotation turns every nonzero direction by $30°$, no direction stays on its own line — and an eigenvector is precisely a direction the matrix leaves on its line. No invariant real direction means no real eigenvector. The geometry of "everything turns" forces the eigenvalues off the real line.

Q2. A real $2\times 2$ matrix has eigenvalue $2 + 3i$. What is its other eigenvalue, and why must this be so?

Answer
The conjugate $2 - 3i$. The conjugate-pair theorem says complex eigenvalues of a real matrix always come in conjugate pairs (conjugating the eigen-equation $A\mathbf{v} = \lambda\mathbf{v}$ leaves the real $A$ unchanged and produces $A\bar{\mathbf{v}} = \bar\lambda\bar{\mathbf{v}}$). Realness of the matrix is the load-bearing condition.

Q3. A complex eigenvalue is written $\lambda = re^{i\theta}$. What does $r$ mean geometrically, and what does $\theta$ mean?

Answer
$\theta$ (the argument) is the angle the transformation rotates by in its invariant plane; $r$ (the modulus) is the factor it scales by. A complex eigenvalue is a rotation-and-scaling instruction: turn by $\theta$, stretch by $r$.

Q4. A real matrix has all its eigenvalues with modulus $r < 1$. What happens to a typical vector under repeated application $\mathbf{x}_{n+1} = A\mathbf{x}_n$?

Answer
It spirals inward to the origin, shrinking by a factor $r$ and rotating by $\theta$ each step — a damped oscillation. The system is stable. Modulus below 1 means the spiral winds in.

Q5. True or false: a complex eigenvector of a real matrix points along a single invariant direction in $\mathbb{R}^2$.

Answer
False. There is no single invariant real direction (that is why the eigenvalue is complex). The complex eigenvector instead packages two real vectors — its real and imaginary parts — that span the invariant plane in which the rotation happens. A complex eigenvector is a plane's frame, not an arrow.

Q6. What is the real canonical (block) form of a real $2\times 2$ matrix with eigenvalues $a \pm bi$, and what transformation does that block perform?

Answer
$C = \begin{psmallmatrix}a & -b\\ b & a\end{psmallmatrix}$, which equals $r\begin{psmallmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{psmallmatrix}$ with $r = \sqrt{a^2+b^2}$, $\theta = \operatorname{atan2}(b,a)$ — a rotation by $\theta$ combined with a scaling by $r$. The block is rotation-scaling in plain real arithmetic.

Q7. For a real $2\times 2$ matrix, how can you tell from the trace and determinant alone whether it has complex eigenvalues?

Answer
Compute the discriminant $\operatorname{tr}(A)^2 - 4\det(A)$. If it is negative, the eigenvalues are a complex conjugate pair (the matrix rotates); if positive, two real eigenvalues; if zero, a repeated real eigenvalue. Negative discriminant = rotation.

Q8. Why must a real $3\times 3$ matrix have at least one real eigenvalue?

Answer
It has three eigenvalues, and complex ones come in conjugate pairs. Three values cannot be partitioned entirely into pairs (an odd count), so at least one must be its own conjugate — i.e., real. Odd dimension forces a leftover real eigenvalue (the fixed axis of a 3D rotation).

Q9. All eigenvalues of a real orthogonal matrix (a rotation/reflection) have what modulus, and why?

Answer
Modulus exactly $1$: an orthogonal matrix preserves length, so it cannot scale any direction — the only allowed scale factor is $1$. Hence every eigenvalue lies on the unit circle ($e^{i\theta}$, or the real values $\pm 1$). Length preservation pins the eigenvalues to the unit circle.

Q10. For a real $2\times 2$ matrix with complex eigenvalues $re^{\pm i\theta}$, what is $\det(A)$ in terms of $r$? Explain geometrically.

Answer
$\det(A) = r^2$. The determinant is the product of the eigenvalues, $\lambda\bar\lambda = |\lambda|^2 = r^2$; geometrically, the transformation scales every length by $r$, so it scales 2D area by $r^2$, and the determinant is exactly the area-scaling factor (Chapter 11). Determinant is the squared modulus because area scales as length squared.

Q11. When you compute the powers $A^n$ of a real matrix with complex eigenvalues via the block form, what happens to the modulus and the angle?

Answer
The modulus is raised to the $n$-th power and the angle is multiplied by $n$: $C^n = r^n\begin{psmallmatrix}\cos n\theta & -\sin n\theta\\ \sin n\theta & \cos n\theta\end{psmallmatrix}$, mirroring $\lambda^n = r^n e^{in\theta}$ (De Moivre). $n$ steps means scale by $r^n$, rotate by $n\theta$.

Q12. What is the connection between the complex eigenvalues of this chapter and the unitary matrices that govern quantum logic gates?

Answer
A unitary matrix ($U^{*}U = I$) is the complex analogue of an orthogonal matrix; like rotations, its eigenvalues all lie on the unit circle. A quantum gate must be unitary so that the squared length of the state (total probability) stays $1$ — it may rotate the state but never grow or shrink it. Complex eigenvalues on the unit circle = reversible, length-preserving rotation, the heart of quantum evolution.