Complex eigenvalues — rotation-scaling interpretation: Question bank

Spot the error

Spot the error: "Since

λ = 1 - i

, the rotation angle is

θ = - 1

(the imaginary part)."

Here

α = 1, β = - 1

, so the angle is

ar g λ = atan2 (β, α) = atan2 (- 1, 1) = - 4 5^{\circ}

, not

β

itself;

β

is a coordinate, the angle is a polar quantity built from both

α

and

β

Spot the error: "

A = (01 - 1 0)

has

det = - 1

, so its eigenvalues are real."

The determinant is

a d - b c = 0 - (- 1) (1) = + 1

, not

- 1

; with

tr = 0

the discriminant is

0 - 4 = - 4 < 0

, giving

λ = \pm i

Spot the error: "The eigenvalues are

λ = 2 \pm i

, so the scale factor is

r = 2

The modulus is

r = 2^{2} + 1^{2} = 5 \approx 2.24

, not the real part; students grab

α

because it is visible, but scale is the distance

∣ λ ∣

Spot the error: "Complex eigenvalues appear because I made an arithmetic mistake — real matrices should have real eigenvalues."

Complex eigenvalues are a genuine, expected outcome whenever the map rotates the plane; a negative discriminant is the correct signal that no real eigen-line exists.

Spot the error: "The block is

(α - β β α)

— I remembered the sign pattern."

The convention (from

λ = α - i β

) puts

- β

in the upper-right and

+ β

in the lower-left: "cos on the diagonal, minus-sin up top," matching a standard rotation matrix un-normalized by

r

Spot the error: "I found

λ = 1 \pm i

, so I write

A = P C P^{- 1}

where

C = (1111)

That

C

has both off-diagonals equal, which is wrong; the rotation-scaling block needs opposite signs off the diagonal,

(11 - 1 1)

Why questions

Why must complex eigenvalues of a real matrix come in conjugate pairs?

Because the Characteristic Polynomial has real coefficients, and taking the complex conjugate of the equation

p (λ) = 0

shows

\overline{λ}

is also a root — so roots pair up as

α \pm i β

Why does splitting

A (x + i y) = (α - i β) (x + i y)

into real and imaginary parts unlock the geometry?

Two complex quantities are equal exactly when real parts match and imaginary parts match; this converts one complex eigen-equation into two real vector equations

A x = α x + β y

and

A y = - β x + α y

that describe how

A

acts on a real basis.

Why is the rotation angle written with

atan2 (β, α)

instead of

arctan (β / α)

Plain

arctan

only returns angles in

(- 9 0^{\circ}, 9 0^{\circ})

and loses the quadrant;

atan2

uses the signs of both

α

and

β

separately, so it places

θ

in the correct quadrant for every combination of signs.

Why does a rotation have no real eigenvector while a stretch does?

A stretch fixes the direction of at least one line (only the length changes), giving a real eigenvector; a rotation turns every arrow off its original line, so no real direction is preserved and the eigen-answer must be complex.

Why is the real part

α

(not the modulus) what controls stability of the ODE

\overset{v}{˙} = A v

The solution scales like

e^{α t}

times an oscillation of frequency

β

; the envelope

e^{α t}

grows only when

α > 0

, so

Re λ

decides growth or decay in continuous time (see Phase Portraits).

Why does the frame

P = (x y)

turn

A

into a clean rotation-scaling even though

A

looked messy?

In the coordinate frame given by the real and imaginary parts of the eigenvector, the two equations

A x = α x + β y

A y = - β x + α y

literally are the columns of the block

C

, so

A = P C P^{- 1}

acts as "rotate then scale" in that frame.

Edge cases

Edge case: What happens when

β = 0

(discriminant

= 0

The pair collapses to a single repeated real eigenvalue

λ = α

; there is no genuine rotation, so the rotation-scaling picture degenerates into a real (possibly defective) case.

Edge case: What is

r

and

θ

for the two roots

λ = \pm i

of the

9 0^{\circ}

rotation?

Both have

r = ∣ \pm i ∣ = 1

(pure rotation, no scaling), but the angles differ by sign: for

λ = + i

θ = atan2 (1, 0) = + 9 0^{\circ}

, while for

λ = - i

θ = atan2 (- 1, 0) = - 9 0^{\circ}

— the conjugate just spins the other way.

Edge case: What if

r = 1

but

θ

is an irrational fraction of

36 0^{\circ}

(using the plain rotation block itself)?

A pure rotation block preserves length, so points stay on a fixed circle and never return exactly to their start, filling the circle densely; the orbit is only an ellipse if

P

is non-orthonormal and pulls the circle into one.

Edge case: What does

α = 0

with

β \neq = 0

mean for the discrete map versus the flow?

Discretely,

∣ λ ∣ = ∣ β ∣

may exceed or fall below 1 and the orbit spirals accordingly; for the flow,

α = 0

gives closed elliptical orbits (a center), showing the two frameworks read the same eigenvalue differently.

Edge case: If

A

is a real symmetric matrix, can it ever have complex eigenvalues?

No — real symmetric matrices always have real eigenvalues (and orthogonal real eigenvectors), so they never rotate; complex eigenvalues require the non-symmetric off-diagonal imbalance seen in rotation-type matrices.

Edge case: What is the smallest matrix size where a real matrix can have complex eigenvalues?

2 \times 2

— a

1 \times 1

real matrix is just a real number and equals its own real eigenvalue, so genuine rotation (and thus complex eigenvalues) first becomes possible in two dimensions.

Recall One-line summary of the traps

Radius vs angle: the modulus $r = ∣ λ ∣$ is the amount of scaling, the argument $θ = ar g λ$ is the amount of spin — never read $α$ or $β$ alone as either. Discrete vs continuous: discrete growth uses $∣ λ ∣$ , continuous growth uses $Re λ = α$ . Pairs: conjugates are forced by real coefficients, and swapping $α + i β$ for $α - i β$ only flips the spin direction. Block layout: $- β$ goes up top ("cos on the diagonal, minus-sin up top").

Question bank — Complex eigenvalues — rotation-scaling interpretation

True or false — justify

Spot the error

Why questions

Edge cases