Complex eigenvalues — rotation-scaling interpretation: Foundations

This page assumes you have seen nothing. Before you can read the parent note Complex eigenvalues — rotation-scaling interpretation, every squiggle it uses must first be built from the ground up. We do that here, one symbol at a time, each resting on the one before.

0. A matrix is a machine that moves points

The picture: a matrix is a movement machine. You put an arrow in, a moved arrow comes out. Do this to every arrow in the plane and the whole plane gets warped — stretched, squished, sheared, flipped, or spun.

Figure — Complex eigenvalues — rotation-scaling interpretation

Two numbers summarise a $2 \times 2$ machine at a glance (full detail in Trace and Determinant):

1. Vectors $v$ , $x$ , $y$ — arrows we can add

The picture: arrows you can slide together and stretch. In the parent note, $x$ and $y$ are not coordinates — they are two whole real arrows that we glue together to build a complex eigenvector. Keep that straight: here $x, y$ name arrows, not axis-positions.

Nonzero matters: the zero arrow $(00)$ has no direction, so it can never be an eigenvector. Every eigenvector definition secretly says "some nonzero $v$ ."

2. The eigen-question: which arrows keep their direction?

The picture below: an eigenvector lies on a line the machine leaves in place (points may slide along that line, but the line itself doesn't tilt). $λ$ says how far along.

3. Reading $det (A - λ I) = 0$ symbol by symbol

To find $λ$ we need three new marks.

Why subtract? The equation $A v = λ v$ rearranges to $A v - λ v = 0$ , i.e. $(A - λ I) v = 0$ : the machine $A - λ I$ crushes the arrow $v$ to zero. A machine can crush a nonzero arrow only if it flattens area to nothing — that is, only if its determinant is $0$ .

The symbol $det (\dots) = 0$ is therefore the sentence: "for which stretch amounts $λ$ does the machine flatten some arrow to nothing?" Those $λ$ are the eigenvalues.

4. The quadratic formula and the discriminant $Δ$

Our equation $λ^{2} - (tr A) λ + det A = 0$ is a quadratic — it has the shape $λ^{2} - p λ + q = 0$ . The tool that solves every quadratic:

That last bullet is why the next section exists.

5. Imaginary unit $i$ and complex numbers $α + i β$

6. Modulus $r$ and argument $θ$ — the polar view

The Cartesian pair $(α, β)$ has a twin description in terms of distance and angle. This is the punchline of the whole topic, so build it carefully on a right triangle.

Drop the point $λ = α + i β$ and connect it to the origin. This makes a right triangle: horizontal leg $α$ , vertical leg $β$ , and the arrow itself as the sloped side (the hypotenuse).

Why $tan$ , and why $atan2$ ? On the triangle, the angle's steepness is captured by $tan θ = \frac{opposite}{adjacent} = \frac{β}{α} .$ To recover $θ$ from $α, β$ we want to undo $tan$ . The plain inverse $arctan (β / α)$ can't tell apart a point and its opposite (both give the same ratio), so it silently fails in half the plane. The fixed-up tool $atan2 (β, α)$ takes the two separate signs of $β$ and $α$ and returns the correct angle in all four quadrants:

$α$	$β$	quadrant of $λ$	$θ = atan2 (β, α)$
$+$	$+$	I (upper right)	between $0°$ and $90°$
$-$	$+$	II (upper left)	between $90°$ and $180°$
$-$	$-$	III (lower left)	between $- 180°$ and $- 90°$
$+$	$-$	IV (lower right)	between $- 90°$ and $0°$
$0$	$+$	straight up	exactly $90°$
$0$	$-$	straight down	exactly $- 90°$

A = (a c b d)

do to an arrow

(y x)

Sends it to

(c x + d y a x + b y)

— it moves/warps the point.

What is

tr A

and

det A

for a

2 \times 2

tr A = a + d

;

det A = a d - b c

What equation defines an eigenvector

v

and eigenvalue

λ

A v = λ v

with

v \neq = 0

— the machine only stretches

v

Why must

v

be nonzero?

The zero arrow has no direction, so it can never mark an eigen-line.

What is the "do-nothing" matrix and how do you build

A - λ I

I = (1001)

; subtract

λ

from each diagonal entry of

A

Why does

det (A - λ I) = 0

find eigenvalues?

(A - λ I)

must crush some nonzero arrow to

0

, which needs zero determinant (zero area).

Write the quadratic formula for

λ

and name

Δ

λ = \frac{tr A \pm Δ}{2}

Δ = (tr A)^{2} - 4 det A

(the discriminant).

Which sign of

Δ

forces complex eigenvalues?

Δ < 0

— the square root is imaginary.

What is

i

, and what is

- 4

i^{2} = - 1

;

- 4 = 2 i

For

λ = α + i β

, which is the real part and which the imaginary part?

α = Re λ

(horizontal),

β = Im λ

(vertical).

What is the conjugate of

α + i β

and why does it appear?

α - i β

; real-coefficient quadratics give complex roots only in mirror pairs.

Define modulus

r

and argument

θ

λ

r = ∣ λ ∣ = α^{2} + β^{2}

(length);

θ = ar g λ

(angle to the horizontal axis).

Why use

atan2 (β, α)

instead of

arctan (β / α)

arctan

can't distinguish a point from its opposite;

atan2

uses both signs to get the correct quadrant.

What does

A = P C P^{- 1}

mean in words?

A

and

C

are the same movement viewed in different coordinate frames (

P

's columns are the new basis arrows).

Write the rotation matrix

R (θ)

and the scaled version

C

R (θ) = (cos θ sin θ - sin θ cos θ)

;

C = r R (θ) = (α β - β α)

Foundations — Complex eigenvalues — rotation-scaling interpretation

0. A matrix is a machine that moves points

1. Vectors $v$ , $x$ , $y$ — arrows we can add

2. The eigen-question: which arrows keep their direction?

3. Reading $det (A - λ I) = 0$ symbol by symbol

4. The quadratic formula and the discriminant $Δ$

5. Imaginary unit $i$ and complex numbers $α + i β$

6. Modulus $r$ and argument $θ$ — the polar view

7. Similarity $A = P C P^{- 1}$ and rotation matrices

Prerequisite map

Equipment checklist