Visual walkthrough — Transpose — definition, properties

2.6.6 · D2 · Maths › Matrices & Determinants — Introduction › Transpose — definition, properties

Parent note Transpose — definition, properties mein paanch properties listed hain. Unme se chaar seedhi-saadhi hain. Lekin ek hai jo deep hai, wahi jo sabse zyada log galat karte hain: product ke liye reversal rule. Yahan hum ise sirf pictures se build karenge — grids, arrows, aur ek careful sum. End tak tumhe dikhega ki order kyun flip hona zaroori hai.

Humein sirf ek purani idea chahiye: ek matrix numbers ka ek grid hota hai, aur do grids ko multiply karne ke liye Matrix Operations ka wikilink. Baaki sab hum yahan build karenge.

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Step 1 — Grid kya hota hai, aur "row $i$, column $j$" ka matlab kya hai

Symbol $[A{]}_{ij}$ ko padha jaata hai "grid $A$ ke row $i$, column $j$ par rehne wala number."

• $A$ ::: numbers ka poora grid
• $i$ ::: kitne boxes neeche (row)
• $j$ ::: kitne boxes across (column)
• $[A{]}_{ij}$ ::: us address par ka single number

Term by term: left side par $i$ ek row hai aur $j$ ek column hai; right side par wahi do labels $A$ ko ulte slots mein diye jaate hain. Yahi ek swap transpose ka poora idea hai.

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Step 2 — Ek grid ko flip karna kaisa dikhta hai

$A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\end{array}\right]\stackrel{\text{flip}}{\to }{A}^T=\left[\begin{array}{cc}{a}_{11}& a_{21}\\ a_{12}& a_{22}\\ a_{13}& a_{23}\end{array}\right]$

Shape badal gayi dekho: ek $2\times 3$ grid ban gayi $3\times 2$ grid. Shape bhi kyun flip hoti hai? Kyunki rows aur columns ne literally apni jobs trade kar li — agar $A$ ke $m$ rows aur $n$ columns the, toh $A^T$ ke $n$ rows aur $m$ columns hain. Ye fact yaad rakho; yahi pehla hint hai ki product rule ka order reverse hona chahiye.

• $a_{11}$ jagah par rehta hai ::: yeh diagonal par hai, iska address $1,1$ swap karne se nahi badlta
• $a_{12}$ aur $a_{21}$ ::: woh diagonal ke across apni jagah trade karte hain

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Step 3 — Multiplication actually kya karta hai (woh tool jo humein chahiye)

Isse pehle ki hum kisi product ka transpose karein, humein crisp hona hoga ki product hota kya hai. Yeh raha ek tool, aur kyun hum yahi use karte hain: multiplication woh akela operation hai jo left grid ki row ko right grid ke column ke saath pair karta hai, aur wahi pairing hai jo transpose scramble karega.

Har symbol padho:

• $[AB{]}_{ij}$ ::: woh number jo hum bana rahe hain, answer ke row $i$, column $j$ par
• $i$ ::: fix karta hai $A$ ki kaunsi row hum slide karenge (figure mein mint highlight)
• $j$ ::: fix karta hai $B$ ka kaunsa column hum slide karenge (coral highlight)
• $k$ ::: running partner index — yeh $1,2,\dots ,p$ walk karta hai, us row ki $k$-th entry ko us column ki $k$-th entry ke saath pair karta hai
• ${\sum }_{k=1}^p$ ::: "woh saare $p$ chhote products add karo"

Aage ke liye crucial observation: $a_{ik}{b}_{kj}$ mein, letter $i$ left factor $a$ par baitha hai, aur letter $j$ right factor $b$ par. In do letters ko dhyaan se dekho.

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Step 4 — WHAT: product ka transpose likh do

Humein $(AB{)}^T$ chahiye. Step 1 ki transpose definition ko single grid $AB$ par apply karo: uske do address labels swap karo.

YE STEP KYU: transpose rule kehta hai $[C^T{]}_{ij}=[C{]}_{ji}$ kisi bhi grid $C$ ke liye; humne bas $C=AB$ choose kiya.

YE KAISA DIKHTA HAI: figure mein, flipped grid ke entry $(i,j)$ ke liye poochhhna matlab hai unflipped grid mein $(j,i)$ par jaana — toh hum $A$ ka row $j$ aur $B$ ka column $i$ pick karte hain.

Right side ke sum ko dhyaan se dekho. Step 3 se compare karo, labels move ho gaye:

• $a_{jk}$ ::: ab $j$ (answer ka column label) $A$ par baitha hai — kyunki humne $A$ ka row $j$ maanga
• $b_{ki}$ ::: ab $i$ (answer ka row label) $B$ par baitha hai
• $k$ ::: abhi bhi running partner, abhi bhi $1$ se $p$ tak sum hota hai

Toh flip ke baad, $i$ $B$ se chipak gaya aur $j$ $A$ se chipak gaya. Ek hi line mein poora mystery hai. Ab hum poochhte hain: kaunsa product naturally exactly yahi pattern produce karta hai?

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Step 5 — WHY: $B^T{A}^T$ build karo aur same sum dekho

Hum guess karte hain answer $B^T{A}^T$ hai (right times left, reversed) aur Step 3 rule se uska $(i,j)$ entry compute karke check karte hain.

$B^T$ ko $A^T$ se us order mein kyun multiply karte hain? Kyunki shapes fit bhi hon iske liye: $A$ tha $m\times p$ aur $B$ tha $p\times n$, toh $A^T$ hai $p\times m$ aur $B^T$ hai $n\times p$. Inhe multiply karne ka ek hi tarika hai $B^T(n\times p)$ times $A^T(p\times m)$ — reversed order hi akela legal order hai. Neeche ki picture mismatched vs matched dimensions dikhati hai.

Ab sum ke andar ke do transposes ko Step 1 se unfliop karo:

• $[B^T{]}_{ik}=b_{ki}$ ::: $B^T$ ke address ke labels swap karo
• $[A^T{]}_{kj}=a_{jk}$ ::: $A^T$ ke address ke labels swap karo

Substitute karo: $[B^T{A}^T{]}_{ij}={\sum }_{k=1}^p{b}_{ki}{a}_{jk}={\sum }_{k=1}^p{a}_{jk}{b}_{ki}$

Last step mein bas do ordinary numbers multiply ho rahe hain jinhe reorder kiya ($b_{ki}{a}_{jk}=a_{jk}{b}_{ki}$) — plain number multiplication order ki parwah nahi karta.

YE KAISA DIKHTA HAI: yeh final sum letter-for-letter identical hai Step 4 ke end mein wale sum se.

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Step 6 — Dono sums same hain, toh rule prove ho gaya

Step 4 ka result Step 5 ke result ke barabar set karo:

$[(AB{)}^T{]}_{ij}={\sum }_{k=1}^p{a}_{jk}{b}_{ki}=[B^T{A}^T{]}_{ij}$

Kyunki yeh har address $(i,j)$ par hold karta hai, toh poore do grids equal hain:

Order reverse hota hai kyunki transposing ne label $i$ ko left factor se right factor par bhej diya, aur $j$ ko right factor se left par. Har letter ko correct grid se attached rakhne ke liye, grids khud sides badalni padi. Symmetric Matrices dekho ek beautiful special case ke liye jahan $A^T=A$ aur kuch flip nahi karna padta.

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Step 7 — Edge cases: kisi corner se surprise mat lo

Ek derivation jise tum stress-test nahi kar sakte, woh derivation jis par tum trust nahi kar sakte. Chaar cases:

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Ek-picture summary

Poora proof do chhote letters ka ek migration hai: $i$ $A$ par start karta hai, $j$ $B$ par start karta hai; flip dono ko trade karta hai; sirf $B^T$ phir $A^T$ hi inhe sahi se pakad sakta hai.

Recall Feynman retelling — kisi dost ko batao jaise

Ek matrix ek grid hai jahan har number ka ek address hota hai: row $i$, column $j$. Transpose karna matlab hai "har address ke dono parts swap karo" — rows tip hokar columns ban jaati hain. Do grids multiply karna matlab hai: left grid se ek row lo, right grid se ek column lo, matching entries multiply karo, add karo. Ab, kisi product ke address $(i,j)$ par entry secretly left grid par $i$ aur right grid par $j$ carry karti hai. Jab tum product transpose karte ho, tum $i$ aur $j$ swap karte ho — toh ab $i$ right grid par hona chahta hai aur $j$ left par. Usse honor karne ke liye, tum dono grids flip karte ho aur unka order swap karte ho: right-wala-transposed pehle, left-wala-transposed baad mein. Isliye $(AB{)}^T=B^T{A}^T$ order reverse karta hai. Addition yeh kabhi nahi karta, kyunki addition kabhi rows ko columns se mix nahi karta — isliye $(A+B{)}^T=A^T+B^T$ order rakhta hai. Simple hai ek pair of dance partners ki tarah jo sides switch karte hain.

Recall Quick self-test

Reversal rule aur uska teen-factor version batao. ::: $(AB{)}^T=B^T{A}^T$ aur $(ABC{)}^T=C^T{B}^T{A}^T$. Order same kyun nahi reh sakta? ::: kyunki sum ${\sum }_k{a}_{jk}{b}_{ki}$ transpose se sirf $B^T{A}^T$ se match karta hai, $A^T{B}^T$ se nahi.

Related paths from here: Determinants ($\det (A^T)=\det A$), Orthogonal Matrices ($A^{T}A=I$), aur Eigenvalues and Eigenvectors jahan $A$ aur $A^T$ ke same eigenvalues hote hain.