Orthogonal matrices — properties, det = ±1: Question bank

4.5.37 · D5 · Maths › Linear Algebra (Full) › Orthogonal matrices — properties, det = ±1

Definition Notation refresher (taaki har symbol samajh mein aaye)

Neeche sab kuch inhi pieces se bana hai — questions se pehle inhe padh lein.

$Q \in R^{n \times n}$ — ek orthogonal matrix hamesha real aur square hoti hai ( $n$ rows, $n$ columns). Yeh optional nahi hai: poori theory ko ek aisi matrix chahiye jo $R^{n}$ ko waapis $R^{n}$ mein map kare taaki uska inverse aur determinant ho sake.
$Q$ ke Columns. $Q = [q_{1} ∣ q_{2} ∣ \dots ∣ q_{n}]$ likhein, jahan $q_{i}$ $i^{th}$ column vector hai — literally $i^{th}$ arrow jo aapko column $i$ seedha neeche padhne se milta hai. $n$ arrows ko side by side khade picture karein.
$Q^{T}$ — transpose: rows aur columns ko flip karein. Numbers ki grid ko uski main diagonal ke across reflect karne ki picture karein.
$I$ — identity matrix: diagonal par 1's, baaki jagah 0's; "kuch nahi karne wala" map.
$q_{i}^{T} q_{j}$ — columns $i$ aur $j$ ka dot product; ek akela number jo bataata hai ki do arrows kitne aligned hain ( $0$ = perpendicular, aur $q_{i}^{T} q_{i} = ∥ q_{i} ∥^{2}$ = arrow ki squared length).
$det Q$ — determinant: woh signed factor jis se map area/volume ko scale karta hai.
Kronecker delta $δ_{ij}$ — shorthand for "1 jab $i = j$ , 0 jab $i \neq = j$ ." Yeh $I$ ki entries ka pattern hai ek ek entry karke likha hua.
Eigenvalue $λ$ aur eigenvector $x$ — ek nonzero arrow $x$ jise $Q$ usi direction mein rakhta hai, sirf rescale karta hai: $Q x = λ x$ . Number $λ$ kitna us special arrow ko stretch kiya gaya. (Deep dive: Eigenvalues and eigenvectors.)
Orthonormal = orthogonal (aapas mein perpendicular) aur normalized (har ek ki length $1$ ). Compactly: $q_{i}^{T} q_{j} = δ_{ij}$ .

True or false — justify

A real matrix whose columns are perpendicular is orthogonal.

False — perpendicular sirf aadhi baat hai; columns ki length bhi 1 honi chahiye. Columns

(0 2), (3 0)

perpendicular hain lekin

Q^{T} Q = diag (4, 9) \neq = I

Every orthogonal matrix has

det = \pm 1

True —

Q^{T} Q = I

det

lene par

(det Q)^{2} = 1

milta hai, isliye

det Q = \pm 1

. Yeh saari orthogonal matrices ki ek genuine property hai.

Every matrix with

det = \pm 1

is orthogonal.

False —

det = \pm 1

zaroori hai, kaafi nahi. Shear

(1051)

det = 1

hai phir bhi space ko skew karta hai; uske columns orthonormal nahi hain.

An orthogonal matrix can be non-square, say

3 \times 2

, as long as its columns are orthonormal.

False — "orthogonal matrix" sirf square

Q \in R^{n \times n}

ke liye define hai; ek

3 \times 2

matrix jiske orthonormal columns hain uske baare mein kaha jaata hai ki uske orthonormal columns hain, lekin uska koi inverse ya determinant nahi hai, isliye woh orthogonal matrix nahi hai.

For an orthogonal matrix,

Q^{- 1} = Q^{T}

True — yahi toh definition ko rearrange karna hai:

Q^{T} Q = I

kehta hai ki

Q^{T}

Q

ko undo karta hai. Isliye orthogonal maps ko invert karna itna sasta hota hai.

The transpose trick

Q^{- 1} = Q^{T}

works for any invertible matrix.

False — yeh tabhi hold karta hai jab

Q

orthogonal ho. Isko trust karne se pehle hamesha

Q^{T} Q = ? I

check karein; Orthogonal matrices — properties, det = ±1 dekhein.

Orthogonal matrices preserve the angle between any two vectors.

True — kyunki

(Q x)^{T} (Q y) = x^{T} y

aur lengths unchanged rehti hain, ratio

cos θ = \frac{x ^{T} y}{∥ x ∥ ∥ y ∥}

same rehta hai, isliye angle survive karta hai — exactly wahi rigidity jo upar figure mein dikhti hai.

An orthogonal matrix can stretch one direction if it shrinks another to compensate.

False — yeh har vector ki length individually preserve karta hai (

∥ Q x ∥ = ∥ x ∥

), isliye koi bhi direction stretch nahi hoti. Sirf volume preservation stretch-and-shrink allow karta; length preservation usse forbid karta hai.

The product of two orthogonal matrices is always orthogonal.

True —

(Q_{1} Q_{2})^{T} (Q_{1} Q_{2}) = Q_{2}^{T} Q_{1}^{T} Q_{1} Q_{2} = Q_{2}^{T} I Q_{2} = I

. Yeh closure hi reason hai ki

O (n)

ek group form karta hai.

The product of two orthogonal matrices always has

det = + 1

False — signs multiply hote hain: do reflections (

- 1

each) dete hain

(- 1) (- 1) = + 1

, lekin ek rotation aur ek reflection milke

- 1

dete hain. Sirf "same sign" pairs

S O (n)

mein land karte hain.

Every orthogonal matrix has real eigenvalues.

False — ek genuine rotation jaise

(c o s θ s i n θ - s i n θ c o s θ)

ke complex eigenvalues

e^{\pm i θ}

hote hain (koi real arrow apni direction nahi rakhta), Eigenvalues and eigenvectors dekhein.

Spot the error

Q

is orthogonal, so

det Q = 1

Determinant

- 1

bhi ho sakta hai. "

= 1

" likhna silently assume karta hai ki rotation hai; reflections bhi orthogonal hoti hain aur unka

det = - 1

hota hai.

Q^{T} Q = I

, therefore

Q Q^{T} = I

need not hold."

Square matrix ke liye dono equivalent hain — ek square matrix ka one-sided inverse automatically two-sided hota hai. Dono automatically saath hold karte hain.

"The columns of

Q

are orthonormal, so its rows might not be."

Ek square orthogonal matrix ki rows bhi orthonormal hoti hain, kyunki

Q Q^{T} = I

exactly wahi rows ke baare mein kehta hai. Columns orthonormal

\Leftrightarrow

rows orthonormal.

det Q = - 1

means

Q

shrinks area, since

- 1 < 1

Magnitude

∣ det Q ∣ = 1

area factor hai, isliye area preserved rehta hai. Minus sign orientation flip (handedness) ke baare mein hai, size ke baare mein nahi — figure ke doosre panel mein dikhaya gaya flip.

"Since

∥ Q x ∥ = ∥ x ∥

, we have

Q x = x

for all

x

Length preserve karna iska matlab nahi ki vector fix ho jaye — ek rotation har nonzero vector ko move karta hai uski length rakhte hue. Equal length

\neq =

equal vector.

A = (1051)

has orthonormal columns because

det = 1

Determinant yahan column length ya perpendicularity ke baare mein kuch nahi kehta: column

(1 5)

ki length

26 \neq = 1

hai aur yeh

(0 1)

se perpendicular nahi hai.

Q

preserves length, so it preserves the dot product of any two vectors — hence it must be the identity."

Dot products preserve karna

Q

ko isometry banata hai, identity nahi. Har rotation aur reflection saare dot products preserve karta hai phir bhi vectors ko move karta hai.

Why questions

Why does

Q^{T} Q = I

force the columns to be orthonormal?

Q^{T} Q

(i, j)

entry columns

i

aur

j

ka dot product

q_{i}^{T} q_{j}

hai; ise

δ_{ij}

(diagonal par 1, baaki jagah 0) ke barabar set karne ka matlab hai ki har column ki length 1 hai (

i = j

) aur alag columns perpendicular hain (

i \neq = j

Why can the determinant only be

\pm 1

and nothing in between, like

0.5

Kyunki

(det Q)^{2} = 1

ke sirf do real solutions hain

\pm 1

; geometrically length preservation saari scaling khatam kar deti hai, isliye

∣ det Q ∣

exactly

1

hona chahiye.

Why must all eigenvalues of

Q

have magnitude

1

Ek eigenvalue

λ

kisi nonzero arrow

x

ke liye

Q x = λ x

satisfy karta hai; tab

∥ x ∥ = ∥ Q x ∥ = ∣ λ ∣ ∥ x ∥

, jo force karta hai

∣ λ ∣ = 1

— map ek eigenvector ki length nahi badal sakta, isliye uska stretch factor unit circle par hota hai.

Why do complex eigenvalues of an orthogonal matrix come in conjugate pairs

e^{\pm i θ}

Q

real hai, isliye uska characteristic polynomial real coefficients wala hai; ek real polynomial ke complex roots hamesha apne conjugates ke saath pair mein aate hain, aur

∣ λ ∣ = 1

unhe

e^{\pm i θ}

par pin karta hai.

Why is

det = + 1

called "rotation" and

- 1

called "reflection"?

det

signed volume factor hai:

+ 1

orientation rakhta hai (spin, figure ka right panel ek spun frame rehta hai),

- 1

ise reverse karta hai (mirror flip), [[Rotations and reflections in

R^{2}

and

R^{3}

]] se link hai.

Why is the transpose, not a full matrix inversion, enough to invert an orthogonal matrix?

Kyunki orthonormal columns directly

Q^{T} Q = I

banate hain, isliye

Q^{T}

hi inverse hai — koi elimination nahi chahiye, yahi cheez QR decomposition ko itna efficient banati hai.

Why does Gram–Schmidt naturally produce the columns of an orthogonal matrix?

Yeh mutually perpendicular unit vectors output karta hai — exactly figure mein cyan arrows — aur unhe columns ke roop mein stack karna exactly condition

Q^{T} Q = I

hai; Orthonormal bases & Gram–Schmidt dekhein.

Edge cases

Is the identity matrix

I

orthogonal, and what is its determinant?

Haan — uske columns standard orthonormal basis hain (figure mein grey arrows), isliye

I^{T} I = I

, aur

det I = + 1

(trivial rotation jo kuch move nahi karta).

Is the negative identity

- I

R^{2}

orthogonal, and is it a rotation or reflection?

Haan, orthogonal;

det (- I) = (- 1)^{2} = + 1

2 \times 2

mein, isliye yeh

18 0^{\circ}

ka rotation hai — reflection nahi, minus sign hone ke bawajood.

- I

R^{3}

a rotation or a reflection?

det (- I) = (- 1)^{3} = - 1

, isliye odd dimensions mein

- I

orientation reverse karta hai aur reflection-type map ke roop mein count hota hai — dimension ki parity decide karti hai.

Can a

1 \times 1

orthogonal matrix exist, and what are the only options?

Haan:

Q = [q]

jahan

q^{2} = 1

, isliye

q = \pm 1

. Sirf do

1 \times 1

orthogonal matrices hain:

[1]

(identity) aur

[- 1]

(line par origin ke through reflection).

Is the zero matrix orthogonal?

Nahi —

0^{T} 0 = 0 \neq = I

, aur uske columns ki length

0

hai,

1

nahi. Ek orthogonal matrix invertible honi chahiye, lekin zero matrix nahi hai.

Can an orthogonal matrix have a zero determinant or be singular?

Kabhi nahi —

det Q = \pm 1 \neq = 0

, isliye har orthogonal matrix invertible hai (inverse

Q^{T}

ke saath). Singularity impossible hai.

Is a permutation matrix (one

1

per row and column) always orthogonal?

Haan — uske columns kisi order mein standard basis vectors hain, isliye orthonormal; determinant

\pm 1

hota hai is baat par depend karte hue ki permutation even hai ya odd number of swaps mein.

Does the set of orthogonal matrices with

det = - 1

form a group?

Nahi — isme identity nahi hai (

det I = + 1

) aur yeh closed nahi hai (unke do ko multiply karne par

det = + 1

milta hai). Sirf

S O (n)

with

det = + 1

ek subgroup hai.