4.5.7
Intuition Bada picture (YE definition KYU hai?)Ek matrix ek machine hai jo vectors ko transform karti hai (ek linear map). Jab tum do matrices A B do machines compose kar rahe ho : "pehle B A hi wajah hai ki row-times-column rule waisa dikhta hai jaisa dikhta hai — yeh ek hi aisa rule hai jo ( A B ) x = A ( B x ) x "matrices composed transformations hain" ke consequences hain.
Definition Matrix productAgar A m × n B n × p C = A B m × p C ij = ∑ k = 1 n A ik B k j . Inner dimensions match karni chahiye (n = n outer dimensions (m , p
Sum-of-products KYU? Chalte hain ise derive karte hain. Ek matrix B x ( B x ) k = ∑ j B k j x j . demand karte hain ki A B A B ( A B ) x = A ( B x ) x i ( A ( B x ) ) i = ∑ k A ik ( B x ) k = ∑ k A ik ( ∑ j B k j x j ) = ∑ j ( = ( A B ) ij k ∑ A ik B k j ) x j .
To x j ( A B ) ij hai — forced hai ki ∑ k A ik B k j Definition arbitrary nahi hai; composition ise force karti hai.
2 × 2 A = ( 1 3 2 4 ) , B = ( 5 7 6 8 ) . ( A B ) 11 = 1 ⋅ 5 + 2 ⋅ 7 = 19 Kyun? A B ( A B ) 12 = 1 ⋅ 6 + 2 ⋅ 8 = 22 Kyun? Row 1 ko column 2 se.
( A B ) 21 = 3 ⋅ 5 + 4 ⋅ 7 = 43 ( A B ) 22 = 3 ⋅ 6 + 4 ⋅ 8 = 50 A B = ( 19 43 22 50 ) .
Worked example Non-square: shapes align karni chahiyeA 2 × 3 B 3 × 2 A = ( 1 − 1 0 3 2 1 ) , B = 4 0 1 1 2 0 . 3 = 3 2 × 2 ( A B ) 11 = 1 ⋅ 4 + 0 ⋅ 0 + 2 ⋅ 1 = 6 Kyun? Shared index k = 1 , 2 , 3 ( A B ) 12 = 1 ⋅ 1 + 0 ⋅ 2 + 2 ⋅ 0 = 1 ( A B ) 21 = − 1 ⋅ 4 + 3 ⋅ 0 + 1 ⋅ 1 = − 3 ( A B ) 22 = − 1 ⋅ 1 + 3 ⋅ 2 + 1 ⋅ 0 = 5 A B = ( 6 − 3 1 5 ) . B A 3 × 3 alag shape . Non-commutativity ka pehla hint!
Worked example Order matters hai (pehle rotation phir reflection ≠ pehle reflection phir rotation)Maano R = ( 0 1 − 1 0 ) F = ( 1 0 0 − 1 ) y R F = ( 0 1 1 0 ) , F R = ( 0 − 1 − 1 0 ) . Yeh step kyun? R F F R R F = F R
Algebraic proof (entrywise). Compatible sizes ke saath,
( ( A B ) C ) i ℓ = ∑ j ( A B ) ij C j ℓ = ∑ j ( ∑ k A ik B k j ) C j ℓ = ∑ k ∑ j A ik B k j C j ℓ . ( A ( B C ) ) i ℓ = ∑ k A ik ( B C ) k ℓ = ∑ k A ik ( ∑ j B k j C j ℓ ) = ∑ k ∑ j A ik B k j C j ℓ . Ho gaya. Associativity hi wajah hai ki "A B C
Yeh bhi note karo ki distributivity hoti hai: A ( B + C ) = A B + A C ( A + B ) C = A C + B C A I = I A = A
Intuition Order kyun matter karta haiMatrices = transformations. Transformations compose karna order-sensitive hai ("pehle moze phir joote" ≠ "pehle joote phir moze"). Sirf special pairs commute karte hain (jaise I
Teen reasons ki A B = B A
Shape: A B B A equal ho hi nahi sakte .Jab dono square bhi hon: dot products rows/cols ko alag tarike se mix karte hain. Upar R F F R Commutator failure measure karta hai: [ A , B ] := A B − B A 0
Worked example Concrete non-commuting square pairA = ( 1 0 1 1 ) B = ( 1 1 0 1 ) A B = ( 2 1 1 1 ) B A = ( 1 1 1 2 ) Kyun? Har entry row·column ke roop mein compute karo; diagonal par alag hain. To [ A , B ] = ( 1 0 0 − 1 ) = 0
Common mistake "Matrix multiplication entrywise hoti hai, jaise addition."Kyun sahi lagta hai: addition hoti hai entrywise, to symmetry suggest karti hai multiplication bhi aisi hogi. Fix: entrywise product (Hadamard) composition ka meaning kho deta hai aur ( A B ) x = A ( B x )
( A B ) − 1 = A − 1 B − 1 Kyun sahi lagta hai: numbers jaisa lagta hai ( ab ) − 1 = a − 1 b − 1 Fix: order reverse hota hai! ( A B ) − 1 = B − 1 A − 1 ( A B ) ( B − 1 A − 1 ) = A ( B B − 1 ) A − 1 = I
A B = 0 ⇒ A = 0 B = 0 Kyun sahi lagta hai: real numbers ke liye true hai (zero divisors nahi hote). Fix: matrices mein zero divisors hote hain, jaise ( 0 0 1 0 ) 2 = 0 0
( A + B ) 2 = A 2 + 2 A B + B 2 Kyun sahi lagta hai: binomial ki aadat. Fix: ( A + B ) 2 = A 2 + A B + B A + B 2 A B + B A 2 A B
Recall Feynman: ek 12-saal ke bachche ko samjhao (hidden)Har matrix ko ek chhoti si recipe machine samjho jo ek grid par dots ko move karti hai. A B B A A row ko B column ke upar slide karo, matching numbers multiply karo aur add karo — yahi row-dot-column rule hai. Kyunki B A A B ( A B ) C A ( B C )
Mnemonic Rules yaad rakho"Rows Cross Columns; Order Counts; Brackets Don't."
RCC = entries ke liye row·column; Order Counts = A B = B A
C = A B C ij ∑ k A ik B k j A i B j
A B Inner wale: A m × n B n × p m × p
Product row·column sums se kyun define hota hai, entrywise nahi? Taaki ( A B ) x = A ( B x ) A B
Associativity state karo aur justify karo. ( A B ) C = A ( B C ) C , B , A ∑ j , k A ik B k j C j ℓ
Kya matrix multiplication commutative hai? Ek counterexample do. Nahi. Jaise ( 1 0 1 1 ) ( 1 1 0 1 ) =
( A B ) − 1 B − 1 A − 1
Matrices ke liye ( A + B ) 2 A 2 + A B + B A + B 2 A 2 + 2 A B + B 2 A , B
Commutator define karo aur yeh kya measure karta hai. [ A , B ] = A B − B A 0 A B
Kya A B = 0 A , B Haan — matrices mein zero divisors hote hain, jaise ( 0 0 1 0 ) 2 = 0
Linear Transformations — matrix product = maps ka composition.Identity and Inverse Matrices — ( A B ) − 1 = B − 1 A − 1 Dot Product — A B Determinants — det ( A B ) = det A det B Transpose — ( A B ) ⊤ = B ⊤ A ⊤ Matrix Powers and Diagonalization — commuting matrices ke eigenvectors share hote hain.
Definition Cij = sum Aik Bkj
Entry / Column / Row views
Associativity AB C = A BC
Non-commutativity AB != BA
Rotation then flip != flip then rotation
