Finding eigenspaces: Question bank — Hinglish deep dive

4.5.30 · D5 · Maths › Linear Algebra (Full) › Finding eigenspaces

True or false — justify karo

True or false: Agar

λ

ek eigenvalue hai toh

A - λ I

invertible hai.

False — ye singular (non-invertible) hona chahiye, kyunki sirf ek singular matrix ke null space mein nonzero vectors hote hain, aur wahi eigenvectors hote hain.

True or false: Eigenspace

E_{λ}

mein zero vector hota hai.

True —

E_{λ} = Null (A - λ I)

ek subspace hai, aur har subspace mein

0

hota hai; lekin

0

ko kabhi eigenvector nahi kehte.

True or false:

λ = 0

kabhi eigenvalue nahi ho sakta.

False —

λ = 0

ka matlab hai

A v = 0

kisi nonzero

v

ke liye, yaani

A

singular hai, aur tab

E_{0} = Null (A)

. Sirf eigenvector nonzero hona chahiye.

True or false: Har

n \times n

matrix mein kam se kam ek real eigenvalue hota hai.

False — ek real rotation matrix (jaise

9 0^{\circ}

rotation) har real vector ko turn kar deta hai, isliye koi real vector apni line par nahi rehta; uske eigenvalues complex hote hain.

True or false: Agar do vectors ek hi

λ

ke eigenvectors hain, toh unka sum bhi eigenvector hai.

True —

E_{λ}

ek subspace hai (ek null space), isliye wo addition ke under closed hai; sum usi "unrotated" set mein rehta hai.

True or false:

λ = 3

ka ek eigenvector

λ = 1

ka bhi eigenvector ho sakta hai.

False — ek nonzero

v

A v = 3 v

aur

A v = v

dono satisfy kare, woh

3 v = v

force karega, jo

v \neq = 0

ke liye impossible hai.

True or false: Ek repeated eigenvalue hamesha 2-dimensional (ya usse bada) eigenspace deta hai.

False — algebraic multiplicity sirf ek upper bound hai;

(5015)

mein

λ = 5

do baar repeat hota hai lekin uska eigenspace sirf ek line hai.

True or false: Geometric multiplicity algebraic multiplicity se zyada ho sakti hai.

False — rule ye hai ki

1 \leq

geometric

\leq

algebraic; eigenspace repeat count se bada kabhi nahi ho sakta.

True or false: Ek diagonalizable matrix mein, har eigenvalue ke liye, geometric multiplicity algebraic multiplicity ke barabar hoti hai.

True — har

λ

ke liye ye equality exactly woh condition hai jisme eigenvectors ek full basis banate hain, aur yahi diagonalizability ka matlab hai.

True or false: Ek eigenvector ko

2

se scale karne par alag eigenvalue milta hai.

False — agar

A v = λ v

hai toh

A (2 v) = λ (2 v)

; scaling se same

λ

rehta hai, kyunki eigenvalue line ki property hai, length ki nahi.

True or false: Ek symmetric matrix deficient ho sakti hai (geometric

<

algebraic).

False — spectral theorem guarantee karta hai ki symmetric matrices hamesha diagonalizable hoti hain, isliye geometric hamesha algebraic ke barabar hoti hai; koi deficiency nahi hoti.

Why questions

Eigenspace dhundhna homogeneous system solve karne jaisa kyun hai?

Kyunki

E_{λ}

define hota hai

Null (A - λ I)

ke roop mein, aur null space dhundna

M v = 0

solve karna hai; eigenvalue

λ

bas batata hai ki konsa matrix

M

use karna hai.

Hum identity

I

kyun daalnate hain taaki

(A - λ I) v

mile, na ki

(A - λ) v

A

ek matrix hai aur

λ

ek scalar hai, isliye

A - λ

undefined hai;

λ v = λ I v

likhne se

λ

ek genuine matrix ban jata hai jise

A

se subtract kiya ja sake.

Eigenspace guaranteed subspace kyun hai, random set kyun nahi?

Ye ek matrix ka null space hai, aur har null space addition aur scalar multiplication ke under closed hota hai aur

0

contain karta hai — ye teen subspace conditions hain.

dim E_{λ} = n - rank (A - λ I)

kyun hai?

Ye Rank-Nullity theorem hai:

n

columns wale matrix ke liye, rank plus nullity

n

ke barabar hota hai, aur nullity exactly eigenspace dimension hai.

Ek matrix diagonalizable kyun nahi ho sakti?

Jab kisi eigenvalue ki geometric multiplicity strictly uski algebraic multiplicity se kam hoti hai, tab independent eigenvectors itne nahi hote ki basis fill ho sake, isliye koi eigenbasis exist nahi karta.

Determinant ko zero set karne se eigenvalues kyun milte hain, eigenvectors kyun nahi?

det (A - λ I) = 0

λ

par ek single scalar condition hai jo kehti hai "ye matrix singular hai"; ye special stretch factors select karta hai, aur sirf ek choose hone ke baad vectors saamne aate hain.

Edge cases

(5005)

ke liye

E_{5}

kya hai, aur kyun?

Pura

R^{2}

— kyunki

A - 5 I

zero matrix hai, har vector

(A - 5 I) v = 0

satisfy karta hai, isliye koi direction special nahi hai.

Agar

A

zero matrix hai, toh uska eigenvalue aur eigenspace kya hai?

Sirf ek eigenvalue hai

λ = 0

(kyunki

A v = 0 = 0 \cdot v

), aur

E_{0} = R^{n}

; har nonzero vector ek eigenvector hai.

Ek upper-triangular matrix ke liye, eigenvalues kahan se padte hain, aur kyun?

Seedha diagonal se, kyunki triangular matrix ka

det (A - λ I)

product

(a_{11} - λ) \dots (a_{nn} - λ)

hota hai, isliye roots exactly diagonal entries hote hain.

Jab

A

ke paas distinct eigenvalues ka full set ho toh eigenspaces ka kya hota hai?

Har eigenvalue ki algebraic multiplicity

1

hoti hai, jo geometric multiplicity

1

force karti hai, isliye har eigenspace ek line hai aur matrix automatically diagonalizable hoti hai.

Kya ek nonzero matrix ke liye eigenspace pura space

R^{n}

ho sakta hai?

Haan — koi bhi scalar-multiple matrix

c I

(jaise

5 I

) mein

E_{c} = R^{n}

hota hai, kyunki wo har vector ko same factor

c

se stretch karta hai bina kisi ko turn kiye.

λ = 0

eigenvalue hona

det A

ke baare mein kya batata hai?

Ye batata hai ki

det A = 0

, kyunki

λ = 0

A - 0 \cdot I = A

singular ho jata hai, aur singular matrices ka determinant zero hota hai.

Connections

Finding eigenspaces — parent recipe jise ye traps stress-test karte hain.
Diagonalization — deficiency questions decide karte hain ki ye kab succeed karta hai.
Symmetric matrices and spectral theorem — "kabhi deficient nahi" ki guarantee.
Rank-Nullity theorem — dimension formula ke peeche yahi hai.

Question bank — Finding eigenspaces

True or false — justify karo

Error pakdo

Why questions

Edge cases

Connections