2.6.6
Intuition Jab hum transpose karte hain toh actually kya ho raha hai?Ek matrix ko ek table ki tarah socho jahan har row ek data record hai. Transpose is table ko flip karta hai taaki rows columns ban jayein aur columns rows. Yeh aisa hai jaise apni spreadsheet ko 90° rotate karke reflect karo. Yeh kyun matter karta hai? Linear algebra mein bahut saare operations (dot products, orthogonality, symmetric matrices) ek matrix ko uske transpose ke saath compare karne par rely karte hain. Yeh ek fundamental symmetry operation hai.
Definition Ek Matrix ka TransposeEk m × n A transpose A T A ′ n × m [ A T ] ij = [ A ] j i Iska matlab: A T i j A j i
Scratch se derivation:
Ek concrete matrix se shuru karte hain:
A = [ a 11 a 21 a 12 a 22 a 13 a 23 ] 2 × 3
Hum "coordinate system" swap karna chahte hain: jo "row i j j i
Step 1: A [ a 11 , a 12 , a 13 ] A T column banna chahiye.
Step 2: A [ a 21 , a 22 , a 23 ] A T column banna chahiye.
Result:
A T = [ a 11 a 13 a 21 a 12 a 23 a 22 ] 3 × 2
Yeh step kyun? Kyunki "row column banti hai" ka matlab hai ki A k A T k
Worked example Basic transposeA = [ 1 4 2 5 3 6 ] A T A [ 1 , 2 , 3 ] A T [ 1 2 3 ] A [ 4 , 5 , 6 ] A T 4 5 6 A T = 1 2 3 4 5 6
Verification: [ A T ] 23 = 6 [ A ] 32 = 6
Worked example Sum ka transposeA = [ 1 3 2 4 ] , B = [ 5 7 6 8 ]
Method 1: Pehle add karo phir transpose karo:
A + B = [ 6 10 8 12 ] ⟹ ( A + B ) T = [ 6 8 10 12 ]
Method 2: Pehle transpose karo phir add karo:
A T = [ 1 2 3 4 ] , B T = [ 5 6 7 8 ] A T + B T = [ 6 8 10 12 ]
Dono match karte hain ✓
Worked example Product ka transposeA = [ 1 3 2 4 ] , B = [ 5 7 6 8 ] Step 1: A B A B = [ 1 ( 5 ) + 2 ( 7 ) 3 ( 5 ) + 4 ( 7 ) 1 ( 6 ) + 2 ( 8 ) 3 ( 6 ) + 4 ( 8 ) ] = [ 19 43 22 50 ] Step 2: A B ( A B ) T = [ 19 22 43 50 ] Step 3: B T A T B T = [ 5 6 7 8 ] , A T = [ 1 2 3 4 ] B T A T = [ 5 ( 1 ) + 7 ( 2 ) 6 ( 1 ) + 8 ( 2 ) 5 ( 3 ) + 7 ( 4 ) 6 ( 3 ) + 8 ( 4 ) ] = [ 19 22 43 50 ] Reversal kyun? Kyunki jab tum ek product ko transpose karte ho, toh "inside" indices jo sum ho rahe the woh positions swap kar lete hain, jisse matrix order flip ho jaata hai.
Definition Symmetric MatrixEk matrix A symmetric hoti hai agar A T = A
Iska matlab: Sabhi i , j a ij = a j i
Example: 1 2 3 2 4 5 3 5 6
Definition Skew-Symmetric MatrixEk matrix A skew-symmetric (ya antisymmetric) hoti hai agar A T = − A Iska matlab: Sabhi i , j a ij = − a j i Zaroori baat: Diagonal elements zero hone chahiye (kyunki a ii = − a ii a ii = 0
Example: 0 − 2 3 2 0 − 5 − 3 5 0
Common mistake Common error: Transpose aur inverse ko confuse karnaGalat soch: "Transpose multiplication ko inverse ki tarah undo karta hai, toh A T = A − 1
Yeh sahi kyun lagta hai: Dono mein kisi na kisi tarah ka "reversal" hota hai — transpose dimensions reverse karta hai (m × n → n × m A A − 1 = I
Sahi baat: Transpose ek structural operation hai (indices flip karna). Inverse ek multiplicative operation hai (A orthogonal matrices kehte hain jahan A T A = I A T = A − 1
Test karo: A = [ 1 2 3 4 ]
A T = [ 1 3 2 4 ] A − 1 = − 2 1 [ 4 − 3 − 2 1 ] = [ − 2 1.5 1 − 0.5 ]
( A B ) T Galat soch: ( A B ) T = A T B T Yeh sahi kyun lagta hai: Sum rule ( A + B ) T = A T + B T
Sahi baat: Matrix multiplication commutative nahi hoti. Jab tum ek product transpose karte ho, summation indices swap ho jaate hain, jisse matrix order reverse ho jaata hai: ( A B ) T = B T A T Mnemonic: "Transpose order reverse karta hai, bilkul ulta padhne ki tarah: ( A B C ) T = C T B T A T
Mnemonic Transpose properties yaad rakhne ke liye"SIRP" chaar core properties ke liye:
S elf-inverse: ( A T ) T = A I gnores scalars: ( k A ) T = k A T R everses products: ( A B ) T = B T A T P reserves sums: ( A + B ) T = A T + B T
Visual: Transpose ko diagonal ke along ek "mirror flip" ki tarah socho. Sums aur scalars ko mirror ki parwah nahi. Products ko parwah hai kyunki jab flip karte ho toh right side wali matrix "pehle mirror se takraati hai".
Recall Ek 12-saal ke bacche ko samjhaoSocho tumhare paas students ki ek data table hai jisme rows mein students hain aur columns mein test scores:
Math Science English
Alice 90 85 88
Bob 75 92 80
Transpose bas is table ko flip kar deta hai taaki students columns ban jayein aur subjects rows:
Alice Bob
Math 90 75
Science 85 92
English 88 80
Yeh kyun karoge? Kabhi kabhi data ek tarah se organize ho toh easy hota hai, kabhi doosre tarah se. Jaise, agar sabhi students ke math scores compare karne hon, toh easy hai jab math ek single row ho jise tum across padh sako.
Cool properties sirf woh rules hain jo batati hain ki tables flip karne par kya hota hai:
Agar do baar flip karo, toh original wapas milta hai (yeh toh obvious hai!)
Agar do tables add karo phir flip karo, toh wahi result milta hai jaise pehle dono tables flip karo phir add karo
Agar do tables multiply karo phir flip karo, toh tables ka order reverse ho jaata hai (yeh thoda tricky hai — yeh isliye hota hai ki multiply karte waqt rows aur columns ek dusre se kaise interact karte hain)
#flashcards/maths
Ek matrix ka transpose kya hota hai? :: Ek m × n A A T n × m [ A T ] ij = [ A ] j i
( A T ) T ( A T ) T = A
Transpose aur addition mein kya relation hai? ( A + B ) T = A T + B T
Transpose aur scalar multiplication mein kya relation hai? ( k A ) T = k A T
Product ( A B ) T ( A B ) T = B T A T
Inverse ( A − 1 ) T ( A − 1 ) T = ( A T ) − 1
Symmetric matrix kya hoti hai? Woh matrix A A T = A a ij = a j i
Skew-symmetric matrix kya hoti hai? Woh matrix A A T = − A a ij = − a j i
( A B ) T = B T A T A T B T Kyunki matrix multiplication mein summation index transpose ke dauran apni position swap kar leta hai, jisse matrix order reverse ho jaata hai.
Agar A 2 × 3 A T A T 3 × 2
rows aur columns flip karo
k-vi row k-vaan column banti hai
Row i col j, row j col i ban jaata hai
Row vectors columns ban jaate hain
Symmetric matrices aur orthogonality
Example check A^T 23 = A 32