Complex eigenvalues — rotation-scaling interpretation

4.5.32 · Maths › Linear Algebra (Full)

Complex eigenvalues kyun aate hain

Ek $2 \times 2$ matrix $A = (a c b d)$ ke liye:

$det (A - λ I) = (a - λ) (d - λ) - b c = λ^{2} - (a + d) λ + (a d - b c)$

Toh $λ^{2} - (tr A) λ + det A = 0$ , jisse milta hai

$λ = \frac{tr A \pm ( tr A ) ^{2} - 4 d e t A}{2} .$

Kya cheez complex banati hai? Discriminant $(tr A)^{2} - 4 det A$ . Agar yeh negative ho, toh roots ek complex conjugate pair $λ = α \pm i β$ hain jahan $β \neq = 0$ .

Ek complex eigenvalue rotation-scaling ko kaise encode karta hai

$λ = α - i β$ lo (convention ke mutabiq yeh sign chunte hain) aur eigenvector $v = x + i y$ lo (jahan $x, y$ real vectors hain). Tab $A v = λ v$ .

Ise expand karte hain. $A (x + i y) = (α - i β) (x + i y)$ .

$A x + i A y = (α x + β y) + i (α y - β x)$

Real aur imaginary parts match karte hain (dono $x, y$ real hain, $A$ real hai, toh $A x, A y$ bhi real hain):

$A x = α x + β y, A y = - β x + α y .$

Ab un dono equations ko columns ki tarah padho. Real matrix $P = (x y)$ banao (columns $x$ aur $y$ ). Tab

$A P = P (α β - β α) .$

Check karo: $A P$ ke columns $A x$ aur $A y$ hain. $P (α β - β α)$ ke columns $α x + β y$ aur $- β x + α y$ hain. ✓ upar se match karta hai.

Toh $A = P C P^{- 1}$ jahan

Yeh factoring kyun kaam karta hai: $(α β - β α)$ form ki har matrix scale $\times$ pure rotation ki tarah factor hoti hai, kyunki $α = r cos θ, β = r sin θ$ . Toh $P$ ke coordinate frame mein (basis vectors $x, y$ ), $A$ aise kaam karta hai jaise " $θ$ se rotate karo, phir $r$ se magnify karo."

Worked Example 1 — pure rotation

$A = (01 - 1 0)$ ( $90°$ rotation matrix).

$tr A = 0, det A = 1$ . Kyun? trace $= a + d = 0$ , det $= a d - b c = 0 - (- 1) = 1$ .
$λ^{2} - 0 \cdot λ + 1 = 0 \Rightarrow λ = \pm i$ . Complex kyun? discriminant $0 - 4 = - 4 < 0$ .
$r = ∣ λ ∣ = 1$ , $θ = ar g (i) ...$ $λ = - i$ lete hain toh $α = 0, β = 1$ , $θ = atan2 (1, 0) = 90°$ .
Interpretation: scale $1$ , rotate $90°$ . Bilkul wahi rotation jo humne shuru mein li thi. ✓

Worked Example 2 — spiral

$A = (11 - 1 1)$ .

$tr = 2, det = 1 \cdot 1 - (- 1) (1) = 2$ . Kyun? standard formulas.
$λ = \frac{2 \pm 4 - 8}{2} = 1 \pm i$ . Yeh kyun? quadratic formula mein daalo; $- 4 = 2 i$ .
$λ = 1 - i$ lo: $α = 1, β = 1$ , toh $r = 2 \approx 1.41$ , $θ = atan2 (1, 1) = 45°$ .
Interpretation: har application $45°$ rotate karta hai aur $2$ se magnify karta hai → ek outward spiral. Bahar kyun? $r > 1$ .

Worked Example 3 — $P$ nikalte hain

Wahi $A = (11 - 1 1)$ , $λ = 1 - i$ . $(A - λ I) v = 0$ solve karo: $A - λ I = (i 1 - 1 i) .$ Row 1: $i v_{1} - v_{2} = 0 \Rightarrow v_{2} = i v_{1}$ . $v_{1} = 1$ lo: $v = (1 i)$ .

$λ = α - i β$ ka eigenvector kyun chuno? Convention taaki $C$ mein lower-left mein $+ β$ aaye; doosra conjugate sirf mirror deta hai.
Split karo: $v = x (10) + i y (01)$ , toh $P = (1001) = I$ .
Tab $C = P^{- 1} A P = A = (11 - 1 1) = 2 (cos 45° sin 45° - sin 45° cos 45°)$ . ✓ Pehle se hi rotation-scaling form mein hai (samajh aata hai — $A$ uss special pattern mein symmetric tha).

Common mistake Steel-manned errors

Galti 1: "Complex eigenvalues matlab koi real eigenvectors nahi, toh $A$ diagonalizable nahi / $R$ pe useless hai." Kyun sahi lagta hai: $R$ pe sach mein koi eigen-line nahi hoti. Fix: $C$ pe woh IS diagonalizable hai; $R$ pe agle best option milta hai — block form $(α β - β α)$ , rotation-scaling. Poori tarah usable hai.

Galti 2: Rotation aur scaling mein confusion. Students $r = α$ , $θ = β$ likhte hain. Kyun sahi lagta hai: $α, β$ toh seedha wahan hain. Fix: $r = α^{2} + β^{2} = ∣ λ ∣$ (modulus = scale), $θ = ar g λ$ (angle = rotation). $α, β$ , $λ$ ke Cartesian coordinates hain, polar nahi.

Galti 3: $θ$ ka sign / spiral ka direction. Eigenvalue $α + i β$ ya $α - i β$ chunna $C$ mein $β$ ka sign palat deta hai, rotation sense ulta ho jaata hai. Fix: yeh sirf basis ki relabeling hai ( $x \leftrightarrow y$ orientation); ek convention chuno ( $λ = α - i β$ ) aur consistent raho.

Galti 4: Sochna ki $r > 1$ ka matlab hamesha "badhta hai." Discrete ke liye $A^{n}$ iterate karte waqt sach hai. ODE $\overset{v}{˙} = A v$ ke liye, growth $Re (λ) = α$ se govern hoti hai, $r$ se nahi. Fix: discrete map → $∣ λ ∣$ use karo; continuous flow → $Re λ$ use karo.

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek record player ghooma rahe ho. Record pe har dot circle mein ghoomta hai — koi bhi ek hi direction mein nahi rehta. Ek "stretch matrix" dots ko seedha fixed lines ke saath bahar dhakelta, lekin ek "spinner" ki koi fixed line hoti hi nahi. Jab math kisi spinner ke liye fixed direction dhundhta hai, toh real wala nahi milta, toh woh "imaginary" numbers se jawab deta hai. Woh imaginary numbers secretly do khabren laate hain: kitna tez ghoomta hai (angle) aur har ghoomne mein kitna bada ya chhota hota hai (size). Toh ek complex eigenvalue bas ek spin-aur-zoom instruction hai disguise mein.

Active Recall

Real 2×2 matrix ke complex eigenvalues kab hote hain?

Jab discriminant

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

ho, yaani characteristic quadratic ke koi real roots nahi hote.

Complex eigenvalues conjugate pairs mein kyun aate hain?

Characteristic polynomial ke real coefficients hote hain, aur ek real polynomial ke complex roots hamesha conjugate pairs

α \pm i β

mein aate hain.

λ = α - i β

ke liye, rotation-scaling ka scale factor kya hai?

r = ∣ λ ∣ = α^{2} + β^{2}

λ = α - i β

ke liye, rotation angle kya hai?

θ = ar g λ = atan2 (β, α)

A

kis real normal form ke similar hai?

A = P C P^{- 1}

jahan

C = (α β - β α) = r (cos θ sin θ - sin θ cos θ)

Complex eigenvector

v = x + i y

P

kaise banate hain?

P = (x y)

— real part

x

pehle column mein, imaginary part

y

doosre column mein.

A v = λ v

se do real equations nikalo jab

λ = α - i β, v = x + i y

ho.

A x = α x + β y

aur

A y = - β x + α y

(real aur imaginary parts match karo).

Discrete iteration

A^{n}

: trajectory bahar ki taraf spiral kab karta hai?

Jab

r = ∣ λ ∣ > 1

. Andar agar

r < 1

, closed ellipse agar

r = 1

Continuous system

\overset{v}{˙} = A v

: growth/decay kya control karta hai?

Re (λ) = α

(modulus nahi):

α > 0

badhta hai,

α < 0

ghatata hai,

α = 0

neutral oscillation.

90°

rotation matrix

(01 - 1 0)

ke eigenvalues?

\pm i

(toh

r = 1, θ = 90°

, pure rotation).

Trace aur determinant eigenvalues ke terms mein 2×2 ke liye?

tr A = λ_{1} + λ_{2} = 2 α

aur

det A = λ_{1} λ_{2} = α^{2} + β^{2} = r^{2}

Connections

Eigenvalues and Eigenvectors — real-eigenvalue baseline jise yeh generalize karta hai.
Characteristic Polynomial — discriminant test ka source.
Rotation Matrices — $r = 1$ special case.
Diagonalization — $C$ pe yeh diagonalizable hain; $R$ pe block form milta hai.
Complex Numbers - Polar Form — modulus/argument = scale/angle.
Linear Dynamical Systems / Phase Portraits — $Re λ$ se spirals vs centers.
Trace and Determinant — quick $α = \frac{tr}{2}$ , $r^{2} = det$ .