Complex eigenvalues — rotation-scaling interpretation: Question bank

4.5.32 · D5 · Maths › Linear Algebra (Full) › Complex eigenvalues — rotation-scaling interpretation

Intuition Is page ko kaise use karein

Har line ek question ::: answer reveal hai. Answer ko cover karo, pehle ek justification socho (sirf true/false nahi), phir check karo. Ye sawaal exactly un jagahon ko target karte hain jahan rotation-scaling picture logon ko confuse karti hai: scale aur angle ko mix karna, sign conventions, aur discrete-vs-continuous growth ka trap.

Shuru karne se pehle, ensure karo ki ye words solid feel hon: trace (diagonal entries ka sum $a + d$ , dekho Trace and Determinant), determinant ( $a d - b c$ ), discriminant (square root ke andar wala part, $(tr A)^{2} - 4 det A$ ), modulus $∣ λ ∣ = α^{2} + β^{2}$ (plane mein origin se $λ$ ki distance), aur argument $ar g λ$ (wo angle jo $λ$ positive real axis ke saath banata hai). Agar inme se kuch bhi shaky lage toh Complex Numbers - Polar Form dekho.

True or false — justify

True or false: Ek real

2 \times 2

matrix jiske complex eigenvalues hain, uske koi real eigenvectors nahi hote.

True — ek complex eigenvalue ka matlab hai koi real number

λ

satisfy nahi karta

A v = λ v

ko, isliye koi real line nahi hai jise map sirf stretch karta ho; har real vector muda jaata hai.

True or false: Agar ek real matrix ka ek complex eigenvalue

α + i β

hai jahan

β \neq = 0

, toh uska doosra, unrelated complex eigenvalue

2 + 5 i

jaisa ho sakta hai.

False — characteristic polynomial ke real coefficients hote hain, isliye complex roots conjugate pairs mein aate hain; doosra eigenvalue forced hota hai

α - i β

hone ke liye.

True or false: Complex eigenvalues ek matrix ko "not diagonalizable" banate hain.

False — ye

C

ke upar diagonalizable hai;

R

ke upar tumhe rotation-scaling block

(α β - β α)

milta hai, jo real-world mein usable form hai (dekho Diagonalization).

True or false: Rotation-scaling ka scale factor per step

α

hota hai.

False —

α

sirf real part hai (

λ

ka ek Cartesian coordinate); scale factor modulus hai

r = α^{2} + β^{2} = ∣ λ ∣

True or false: Discrete map

v \mapsto A v

ke iteration ke liye,

∣ λ ∣ > 1

ka matlab hai orbits baahir ki taraf spiral karti hain.

True — har step origin se distance ko

r = ∣ λ ∣

se multiply karta hai, isliye

r > 1

radius badhata hai aur outward spiral produce karta hai (dekho Linear Dynamical Systems).

True or false: Continuous flow

\overset{v}{˙} = A v

ke liye,

∣ λ ∣ > 1

ka matlab hai trajectories blow up ho jaati hain.

False — ODE ke liye, growth

Re λ = α

se set hoti hai;

α > 0

grow karta hai,

α < 0

decay karta hai,

α = 0

closed orbit hai,

∣ λ ∣

ki parwah kiye bina.

True or false: Ek pure rotation matrix (koi scaling nahi) ke eigenvalues unit circle par hote hain.

True — pure rotation ka matlab

r = ∣ λ ∣ = 1

, isliye

λ = cos θ \pm i sin θ

exactly unit circle par baithta hai (dekho Rotation Matrices).

True or false: Agar

det A < 0

, toh eigenvalues phir bhi complex ho sakte hain.

False — complex eigenvalues ke liye

(tr A)^{2} - 4 det A < 0

chahiye, jo force karta hai

det A > (tr A)^{2} /4 \geq 0

; ek negative determinant guarantee karta hai opposite sign ke do real eigenvalues.

True or false: Eigenvalue

α + i β

choose karna instead of

α - i β

actual matrix

A

ko change karta hai.

False — ye sirf

P

ke basis ko relabel karta hai (

x

aur

y

ke roles swap karta hai), block mein

β

ka sign flip karta hai aur isliye visual spin direction, lekin

A

khud unchanged rehta hai.

Spot the error

Spot the error: "Kyunki

λ = 1 - i

hai, rotation angle

θ = - 1

hai (imaginary part)."

Yahan

α = 1, β = - 1

hai, isliye angle

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

hai,

β

nahi;

β

ek coordinate hai, angle ek polar quantity hai jo dono

α

aur

β

se banta hai.

Spot the error: "

A = (01 - 1 0)

det = - 1

hai, isliye iske eigenvalues real hain."

Determinant hai

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

- 1

nahi;

tr = 0

ke saath discriminant hai

0 - 4 = - 4 < 0

, jisse

λ = \pm i

milta hai.

Spot the error: "Eigenvalues hain

λ = 2 \pm i

, isliye scale factor

r = 2

hai."

Modulus hai

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

, real part nahi; students

α

pakad lete hain kyunki wo visible hai, lekin scale distance

∣ λ ∣

hai.

Spot the error: "Complex eigenvalues isliye aaye kyunki maine arithmetic mistake ki — real matrices ke real eigenvalues hone chahiye."

Complex eigenvalues ek genuine, expected outcome hai jab bhi map plane ko rotate karta hai; ek negative discriminant correct signal hai ki koi real eigen-line exist nahi karti.

Spot the error: "Block hai

(α - β β α)

— maine sign pattern yaad rakha."

Convention (

λ = α - i β

se) upper-right mein

- β

aur lower-left mein

+ β

rakhta hai: "diagonal par cos, upar minus-sin," jo ek standard rotation matrix se match karta hai

r

se un-normalized.

Spot the error: "Maine

λ = 1 \pm i

find kiya, toh main likhta hoon

A = P C P^{- 1}

jahan

C = (1111)

C

ke dono off-diagonals equal hain, jo galat hai; rotation-scaling block ko off-diagonal par opposite signs chahiye,

(11 - 1 1)

Why questions

Kyun real matrix ke complex eigenvalues conjugate pairs mein aane chahiye?

Kyunki Characteristic Polynomial ke real coefficients hote hain, aur equation

p (λ) = 0

ka complex conjugate lene par pata chalta hai ki

\overline{λ}

bhi ek root hai — isliye roots

α \pm i β

ke roop mein pair up hoti hain.

Kyun

A (x + i y) = (α - i β) (x + i y)

ko real aur imaginary parts mein split karna geometry unlock karta hai?

Do complex quantities exactly tab equal hote hain jab real parts match karein aur imaginary parts match karein; ye ek complex eigen-equation ko do real vector equations

A x = α x + β y

aur

A y = - β x + α y

mein convert karta hai jo describe karte hain ki

A

ek real basis par kaise act karta hai.

Kyun rotation angle ko

atan2 (β, α)

ke saath likha jaata hai instead of

arctan (β / α)

ke?

Plain

arctan

sirf

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

mein angles return karta hai aur quadrant kho deta hai;

atan2

dono

α

aur

β

ke signs alag se use karta hai, isliye ye

θ

ko har sign combination ke liye correct quadrant mein rakhta hai.

Kyun rotation ka koi real eigenvector nahi hota jabki stretch ka hota hai?

Ek stretch kam se kam ek line ki direction fix rakhta hai (sirf length badlti hai), ek real eigenvector deta hai; rotation har arrow ko uski original line se hatata hai, isliye koi real direction preserve nahi hoti aur eigen-answer complex hona chahiye.

Kyun real part

α

(modulus nahi) ODE

\overset{v}{˙} = A v

ki stability control karta hai?

Solution

e^{α t}

times frequency

β

ka oscillation ki tarah scale hota hai; envelope

e^{α t}

sirf tab grow karta hai jab

α > 0

ho, isliye

Re λ

continuous time mein growth ya decay decide karta hai (dekho Phase Portraits).

Kyun frame

P = (x y)

A

ko ek clean rotation-scaling mein turn kar deta hai, jabki

A

messy lag rha tha?

Eigenvector ke real aur imaginary parts se diye gaye coordinate frame mein, do equations

A x = α x + β y

A y = - β x + α y

literally block

C

ke columns hain, isliye

A = P C P^{- 1}

us frame mein "rotate then scale" ki tarah act karta hai.

Edge cases

Edge case: Kya hota hai jab

β = 0

(discriminant

= 0

Pair ek single repeated real eigenvalue

λ = α

mein collapse ho jaata hai; koi genuine rotation nahi hai, isliye rotation-scaling picture ek real (possibly defective) case mein degenerate ho jaati hai.

Edge case:

9 0^{\circ}

rotation ke do roots

λ = \pm i

ke liye

r

aur

θ

kya hai?

Dono ka

r = ∣ \pm i ∣ = 1

hai (pure rotation, koi scaling nahi), lekin angles sign se differ karte hain:

λ = + i

ke liye,

θ = atan2 (1, 0) = + 9 0^{\circ}

, jabki

λ = - i

ke liye,

θ = atan2 (- 1, 0) = - 9 0^{\circ}

— conjugate sirf doosri taraf spin karta hai.

Edge case: Kya hoga agar

r = 1

ho lekin

θ

36 0^{\circ}

ka ek irrational fraction ho (plain rotation block hi use karke)?

Ek pure rotation block length preserve karta hai, isliye points ek fixed circle par rehte hain aur exactly apne start par kabhi nahi laute, circle ko densely fill karte hain; orbit sirf tab ek ellipse hoti hai jab

P

non-orthonormal ho aur circle ko ek mein kheench le.

Edge case: Discrete map versus flow ke liye

α = 0

with

β \neq = 0

ka kya matlab hai?

Discretely,

∣ λ ∣ = ∣ β ∣

1 se upar ya neeche ho sakta hai aur orbit accordingly spiral karta hai; flow ke liye,

α = 0

closed elliptical orbits deta hai (ek center), jo dikhata hai ki dono frameworks ek hi eigenvalue ko alag tarah padhte hain.

Edge case: Agar

A

ek real symmetric matrix hai, kya uske kabhi complex eigenvalues ho sakte hain?

Nahi — real symmetric matrices ke hamesha real eigenvalues hote hain (aur orthogonal real eigenvectors), isliye ye kabhi rotate nahi karte; complex eigenvalues ko rotation-type matrices mein dekhe jaane wale non-symmetric off-diagonal imbalance ki zaroorat hoti hai.

Edge case: Smallest matrix size kya hai jahan ek real matrix ke complex eigenvalues ho sakte hain?

2 \times 2

— ek

1 \times 1

real matrix sirf ek real number hai aur apna khud ka real eigenvalue hai, isliye genuine rotation (aur thus complex eigenvalues) pehli baar do dimensions mein possible hoti hai.

Recall Traps ki one-line summary

Radius vs angle: modulus $r = ∣ λ ∣$ scaling ki amount hai, argument $θ = ar g λ$ spin ki amount hai — kabhi bhi $α$ ya $β$ ko akele dono mein se kuch bhi na padho. Discrete vs continuous: discrete growth $∣ λ ∣$ use karta hai, continuous growth $Re λ = α$ use karta hai. Pairs: conjugates real coefficients se forced hain, aur $α + i β$ se $α - i β$ swap karna sirf spin direction flip karta hai. Block layout: $- β$ upar jaata hai ("diagonal par cos, upar minus-sin").