4.5.37
Ek orthogonal matrix basically ek rigid motion (rotation ya reflection) ka matrix version hai.
Yeh vectors ko move karta hai lekin unhe kabhi stretch, shrink, ya skew nahi karta . Lengths waisi hi rehti hain,
angles waisi hi rehti hain. Kyunki volume preserve hota hai, determinant ko "no scaling" measure karna padta hai —
jo force karta hai ki det = ± 1 orientation : + 1 − 1
Definition Orthogonal matrixEk real square matrix Q ∈ R n × n orthogonal hai agar
Q T Q = Q Q T = I . Q Q − 1 = Q T orthogonal group O ( n )
YEH definition KYO? Condition Q T Q = I Q orthonormal set form karte hain (mutually perpendicular unit vectors). Dekho kyun.
Q T Q = I Q = [ q 1 ∣ q 2 ∣ ⋯ ∣ q n ] q i Q T Q ( i , j ) dot product q i T q j Q T Q = I q i T q j = δ ij = { 1 0 i = j i = j i = j i = j
Yeh defining geometric property hai — isse scratch se derive karo.
Consequences (HOW yeh geometry deta hai):
Length: y = x ∥ Q x ∥ 2 = ( Q x ) T ( Q x ) = x T x = ∥ x ∥ 2 ∥ Q x ∥ = ∥ x ∥ Angle: kyunki cos θ = ∥ x ∥ ∥ y ∥ x T y
Toh Q isometry hai: ek distance-preserving linear map.
Intuition Sign ka matlab kya haidet Q ∣ det Q ∣ = 1 Sign orientation record karta hai:
det Q = + 1 special orthogonal Q ∈ S O ( n ) rotation (right hand, right hand hi rehta hai).det Q = − 1 reflection (ya rotation-times-reflection) — handedness flip ho jaati hai.
det Q = ± 1 har matrix jinka det = ± 1 Kyun sahi lagta hai: property det = ± 1 Kyun galat hai: det = ± 1 necessary hai lekin sufficient nahi . Example:
A = ( 1 0 5 1 ) det = 1 A T A = I Fix: orthogonality poori condition hai Q T Q = I det = ± 1 byproduct hai.
Intuition Rows bhi orthonormal hoti hainKyunki Q Q T = I Q rows bhi ek orthonormal set form karti hain. Columns
orthonormal ⇔
Worked example 4 — Forecast-then-VerifyForecast: kya P = ( 0 1 1 0 ) det Verify: columns e 2 , e 1 det = 0 ⋅ 0 − 1 ⋅ 1 = − 1 y = x − 1
Recall Feynman: ek 12-saal ke bacche ko samjhaoEk flat sheet stickers ki imagine karo. Ek orthogonal matrix us sheet ko spin karne ya ise pancake ki tarah palat dene ka tarika hai —
lekin kabhi stretch ya squish nahi karna. Har sticker same size rehta hai, aur
kisi bhi do stickers ke beech ki distance kabhi nahi badlti. Agar tum sirf spin karo, toh tum ise waapis rakh sakte ho
same tarah — woh "+ 1 flip karna padta hai, woh "− 1 + 1 − 1
"O for Orthonormal, T for Transpose-is-inverse, ±1 because flip-or-spin."
Q T Q = I R otation = R ight-hand kept = + 1 L ection fL ips = − 1
Common mistake "Orthogonal matrix matlab entries orthogonal-looking hain / bahut saare zeros hain."Sahi lagta hai: swap aur identity matrices sparse hoti hain.
Galat: 5 1 3 4 − 4 3 Fix: "orthogonal" refer karta hai columns ke orthonormal vectors hone se, zero entries se nahi.
Q − 1 = Q T Sahi lagta hai: transpose sasta hai, ek shortcut inverse jaisa lagta hai.
Galat: sirf tab hi hold karta hai jab Q Fix: shortcut trust karne se pehle Q T Q = ? I
Ek orthogonal matrix ki definition? Ek real square Q Q T Q = Q Q T = I Q − 1 = Q T
Q T Q = I Woh orthonormal hain: unit length aur mutually perpendicular (q i T q j = δ ij
det Q = ± 1 det ( Q T Q ) = det I ⇒ ( det Q ) 2 = 1 ⇒ det Q = ± 1
det Q = + 1 − 1 + 1 S O ( n ) − 1
∥ Q x ∥ = ∥ x ∥ ∥ Q x ∥ 2 = x T Q T Q x = x T x = ∥ x ∥ 2
Kya det = ± 1 Nahi — necessary hai sufficient nahi (e.g. ek shear). Poori condition Q T Q = I
Q Saare eigenvalues ∣ λ ∣ = 1 ± 1 e ± i θ
Ek orthogonal matrix ka inverse? Q − 1 = Q T
Kya do orthogonal matrices ka product orthogonal hota hai? Haan — O ( n ) ( Q 1 Q 2 ) T Q 1 Q 2 = I
2 × 2 det cos 2 θ + sin 2 θ = + 1
Orthonormal bases & Gram–Schmidt — Q build karna.Determinants — properties — det ( A B ) = det A det B det A T = det A Eigenvalues and eigenvectors — ∣ λ ∣ = 1 Rotations and reflections in $\mathbb{R}^2$ and $\mathbb{R}^3$ — geometric instances.QR decomposition — A = QR Q Spectral theorem — symmetric matrices ko orthogonal Q Inner product spaces — isometry / length preservation generalized.
