1.1.8
Ek matrix A transformation hai jo space ko stretch, rotate, aur shear karti hai. Iska inverse A − 1 A bilkul undo kar deta hai — yeh output ko wapas input pe le jaata hai.
ML mein WHY matters: A x = b ek unique solution hota hai vs. koi nahi ya infinitely many . Invertibility ka mathematical naam hai "yeh transformation koi information khoता nahi."
Ek square n × n A A − 1 A A − 1 = A − 1 A = I n I n A invertible (ya nonsingular ) kehte hain. Agar exist nahi karta, toh A singular hai.
A A − 1 n n pseudo-inverse exist karta hai).
Hum formula memorize nahi karte — hum use build karte hain. Maano
A = [ a c b d ] , A − 1 = [ x z y w ] .
Hum demand karte hain A A − 1 = I [ a c b d ] [ x z y w ] = [ 1 0 0 1 ] .
Yeh step kyun? Inverse ki definition exactly yahi equation hai — hum bas ise likhte hain aur solve karte hain.
Column 1 deta hai: a x + b z = 1 c x + d z = 0 z = − d c x a x − d b c x = 1 ⟹ x d a d − b c = 1 ⟹ x = a d − b c d .
Yeh step kyun? Hum x a d − b c khud bahar aata hai — woh quantity hai jo divide karne ke liye nonzero honi chahiye.
Baaki charon unknowns ke liye bhi yahi karo:
Poori cheez fail ho jaati hai agar a d − b c = 0 determinant hai, aur yeh invertibility ka gatekeeper hai.
Intuition Determinant = signed volume scale factordet A A det A = 0 collapse kar deta hai. Ek baar volume zero ho jaaye, toh use recover nahi kar sakte — infinitely many inputs ek hi output pe map ho jaate hain, isliye koi unique undo exist nahi karta . Isliye A − 1 iff det A = 0
WHY #7? Agar A v = 0 ⋅ v v = 0 0 det A = ∏ λ i det A = 0
WHY yeh kaam karta hai: identity A ⋅ adj ( A ) = ( det A ) I det A n ≥ 3 Gauss–Jordan use karte hain ([ A ∣ I ] → [ I ∣ A − 1 ]
Worked example Example 1 — ek clean2 × 2 A = [ 4 2 7 6 ] Step 1: det A = 4 ⋅ 6 − 7 ⋅ 2 = 24 − 14 = 10 Kyun? Nonzero → invertible.
Step 2: Diagonal swap karo, off-diagonal negate karo: [ 6 − 2 − 7 4 ] Kyun? Yeh 2 × 2 Step 3: 10 A − 1 = [ 0.6 − 0.2 − 0.7 0.4 ] Check: A A − 1 = I
Worked example Example 2 — ek singular matrixB = [ 1 2 2 4 ] Step 1: det B = 1 ⋅ 4 − 2 ⋅ 2 = 0 Kyun matters? Zero determinant → not invertible .
Observe: row 2 = 2 × = 1 < 2 B x = 0 x = [ 2 − 1 ] [ 1 ⋅ 2 + 2 ( − 1 ) 2 ⋅ 2 + 4 ( − 1 ) ] = 0
Worked example Example 3 — inverse se solve karnaA x = b A b = [ 1 1 ] x = A − 1 b = [ 0.6 − 0.2 − 0.7 0.4 ] [ 1 1 ] = [ − 0.1 0.2 ] Unique kyun? Kyunki det A = 0 (ML mein hum rarely A − 1
( A B ) − 1 = A − 1 B − 1 Kyun sahi lagta hai: scalars ke liye products pe distribute karna kaam karta hai aur transposes bhi similar lagte hain. Fix: inverses order reverse karte hain jaise kapde pehanna/utaarna:
( A B ) − 1 = B − 1 A − 1 . ( A B ) ( B − 1 A − 1 ) = A ( B B − 1 ) A − 1 = A I A − 1 = I
Common mistake "Sirf square matrices matter karti hain, isliye har square matrix invertible hoti hai."Kyun sahi lagta hai: inverse sirf square matrices ke liye defined hai, toh "square ⇒ invertible" automatic lagta hai. Fix: square hona necessary hai lekin sufficient nahi . [ 1 2 2 4 ] det = 0 bhi full rank / nonzero determinant chahiye.
Common mistake "Ek chota nonzero determinant theek hai."Kyun sahi lagta hai: technically det = 0 Fix: numerically, ek nearly-singular matrix (bada condition number ) errors ko wildly amplify karta hai. ML mein hum regularization add karte hain (A + λ I
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho ek machine hai jo photo leti hai aur use ek fixed tarike se scramble karti hai. Inverse machine use wapas original pe unscramble karti hai. Lekin agar scrambler kabhi picture ka kuch hissa throw away kar de (jaise uska aadha hissa paint kar do), toh koi un-scrambler use wapas nahi la sakta — bahut kuch kho gaya. Determinant ek number hai jo batata hai "kya machine ne kuch throw away kiya?" Agar yeh zero hai, toh information kho gayi aur koi undo button nahi hai . Agar zero nahi hai, toh tum hamesha use perfectly undo kar sakte ho.
Mnemonic Conditions yaad karo"Dr. Rank Loves Nulls Uniquely, Not Eigen-zeros, Ident-ifiable."
D et≠0 · Rank full · L inearly independent cols · Null space trivial · U nique solutions · Not 0 eigenvalue · row-reduces to Ident ity. Sab saath uthte hain ya saath girte hain.
Inverse A − 1 A A − 1 = A − 1 A = I
Ek 2 × 2 a d − b c 1 [ d − c − b a ]
Ek matrix invertible hai iff uska determinant...? nonzero ho (det A = 0
Invertibility ke equivalent teen conditions batao. Full rank, linearly independent columns, trivial null space (bhi: unique solutions, 0 eigenvalue nahi, row-reduces to I
( A B ) − 1 B − 1 A − 1
Ek non-square matrix ka true inverse kyun nahi ho sakta? Yeh dimension change karta hai, isliye koi single matrix outputs ko one-to-one wapas map nahi kar sakti; sirf pseudo-inverse exist karta hai.
Agar det A = 0 Space ko ek lower dimension pe collapse karta hai, information kho jaati hai → koi unique undo nahi.
Determinant aur eigenvalues ka relation? det A = ∏ i λ i det = 0
Chota determinant numerically risky kyun hai? Bada condition number rounding errors amplify karta hai; matrix nearly singular hoti hai (fix: + λ I
adj ( A ) Cofactor matrix ka transpose; A − 1 = d e t A 1 adj ( A )
Matrix A as transformation
Swap diag negate off divide by det
Unique solution to Ax = b
Linear regression normal equations
