Matrix multiplication — definition, associativity, non-commutativity: Visual walkthrough

4.5.7 · D2 · Maths › Linear Algebra (Full) › Matrix multiplication — definition, associativity, non-commu

Pehli line se pehle, teen seedhe words jo hum bar bar use karenge:

Baaki sab neeche earn hoga.

Step 1 — Ek matrix ek vector khaati hai aur ek vector ugalti hai

KYA. Ek $2 \times 2$ matrix $B$ aur ek vector $x$ ke numbers lo. Rule " $B$ acting on $x$ ", likha jaata hai $B x$ , ek naya vector produce karta hai. Uska $k$ -th slot hai: $(B x)_{k} = row k, col 1 B_{k 1} 1st slot of x x_{1} + row k, col 2 B_{k 2} 2nd slot x_{2} = \sum_{j} B_{k j} x_{j} .$

Annotation padho: output slot $k$ paane ke liye, tum $B$ ki row $k$ ke saath chalte ho, aur har number ko $x$ ke matching slot ke saath pair karte ho, multiply karte ho, aur add karte ho. Symbol $\sum_{j}$ ("sum over $j$ ") bas shorthand hai "har column $j$ ke liye yeh karo aur results add karo."

YEH HI KYUN, kuch simple kyun nahi? Kyunki hum insist karte hain ki $B$ ek linear machine ho: use grid ko ek naye grid mein bhejna hoga jahan straight lines straight rahen aur evenly-spaced dots evenly spaced rahen. Isi property wala formula sirf yahi weighted sum hai. (Koi bhi curved ya lopsided rule grid ko bend kar deta.)

PICTURE. Grid ke basis arrows $\overset{e}{^}_{1} = (10)$ (right) aur $\overset{e}{^}_{2} = (01)$ (up) $B$ ke do columns par bhej diye jaate hain. Koi bhi $x$ saath mein carry ho jaata hai.

Figure — Matrix multiplication — definition, associativity, non-commutativity

Step 2 — Do machines back to back: composition ka sawaal

KYA. Ab do machines ek ke baad ek chalao: pehle $B$ , phir $A$ . $x$ se shuru karo, $B x$ pao, phir woh $A$ mein daalo aur $A (B x)$ pao. Brackets ka matlab hai "andar wala pehle."

KYUN. Yahi wajah hai matrix multiplication exist karti hai. Hum ek single machine $C$ chahte hain taaki $C$ ko ek baar use karna " $B$ then $A$ " ke barabar ho: $C x = A (B x) for every x .$ Us single machine ka naam hai $C = A B$ . Reading order note karo: machine jo right par likhi hai ( $B$ ) pehle act karti hai, kyunki woh $x$ ke sabse kareeb hai. Yahi right-to-left habit hai jahan se baad mein har "order reverses" wala surprise aata hai.

PICTURE. Ek dot do boxes se flow karta hai: grid $\to$ (box $B$ ) $\to$ tilted grid $\to$ (box $A$ ) $\to$ final grid.

Step 3 — Composition promise se formula nikalna

KYA. Hum $A (B x)$ slot by slot compute karte hain aur padh lete hain ki combined machine ki entries kya honi chahiye. Output slot $i$ lo: $(A (B x))_{i} = \sum_{k} A_{ik} (B x)_{k} = \sum_{k} A_{ik} (\sum_{j} B_{k j} x_{j}) .$

Term by term: $A_{ik}$ = $A$ ki row $i$ , entry $k$ . $(B x)_{k}$ = intermediate vector ka slot $k$ , jise hum Step 1 mein already expand kar chuke hain. Toh $k$ shared index hai — woh slot jahan $A$ ke columns $B$ ki rows se milte hain.

Ab do sums ko swap karo aur har $x_{j}$ ke aage wala coefficient collect karo: $= \sum_{j} (coefficient of x_{j} k \sum A_{ik} B_{k j}) x_{j} .$

SWAP LEGAL KYUN HAI. Yeh ordinary numbers ke finite sums hain, isliye reorder karna kabhi total nahi badalta (yeh bas addition rearranged hai). Koi calculus nahi, koi limits nahi — bas $a + b = b + a$ kai baar apply kiya.

The punchline. Single machine $C = A B$ ke liye hum yeh bhi chahte hain ki $(C x)_{i} = \sum_{j} C_{ij} x_{j}$ . Dono expressions mein $x_{j}$ ka coefficient match karne par force hota hai: $C_{ij} = k \sum A_{ik} B_{k j} .$ Kuch choose nahi kiya gaya. "Compose" shabd ne yeh equation likhi.

PICTURE. Bracketed coefficient literally $A$ ki row $i$ , $B$ ke column $j$ par slide karke hai: line them up, multiply pairs, add.

Step 4 — Ek entry bante dekho (numbers ka close-up)

KYA. Parent ke example ke saath concretely: $A = (1324)$ , $B = (5768)$ , top-left entry compute karo: $(A B)_{11} = 1 A_{11} 5 B_{11} + 2 A_{12} 7 B_{21} = 5 + 14 = 19.$

Index bookkeeping: $i = 1$ ( $A$ ki top row), $j = 1$ ( $B$ ka left column), aur $k$ us row aur column ke andar $1, 2$ tak run karta hai.

SIRF EK ENTRY KYUN DIKHAO? Kyunki pattern hi lesson hai — har doosri entry same dance hai alag $i, j$ ke saath. Chaar baar karo aur parent wala $(19432250)$ mil jaata hai.

PICTURE. $A$ ki row 1 (horizontal) aur $B$ ka column 1 (vertical) highlight karo; unka crossing $A B$ ke top-left slot ko feed karta hai.

Step 5 — Order counts: wohi do machines, swap karo

KYA. $R = (01 - 1 0)$ lo (90° left turn) aur $F = (10 0 - 1)$ lo (flip up↔down). Tab $R F = (0110) (flip, then turn), F R = (0 - 1 - 1 0) (turn, then flip) .$

YEH DIFFER KYUN KARTE HAIN. Step 2 yaad karo: right wali machine pehle act karti hai. Point $(10)$ track karo: flip-then-turn usse upar le jaata hai; turn-then-flip usse neeche le jaata hai. Alag final spot ⟹ alag machines ⟹ $R F \neq = F R$ . Yahi non-commutativity hai, aur yeh normal case hai, exception nahi.

PICTURE. Wohi chhota sa flag do boxes mein dono orders mein push kiya — do alag jagah end hota hai.

Step 6 — Degenerate aur edge cases (koi scenario nahi chhoota)

KYA. Teen cases jo logon ko trip karte hain:

Shapes lock hone chahiye. Agar $A$ $2 \times 3$ hai aur $B$ $3 \times 2$ hai, toh inner dimensions ( $3 = 3$ ) touch karke vanish ho jaate hain; outer wale ( $2, 2$ ) bachte hain: $A B$ $2 \times 2$ hai. Lekin $B A$ $3 \times 3$ hai — alag shape, toh usse $A B$ se compare bhi nahi kar sakte. Mismatched inner dimensions ( $2 \times 3$ times $2 \times 3$ ) simply undefined hain — row aur column ki lengths alag hain, toh dot product form hi nahi ho sakta.
Identity $I$ . $I = (1001)$ grid ko untouched chhodta hai: $A I = I A = A$ . Iske columns $\overset{e}{^}_{1}, \overset{e}{^}_{2}$ hain, isliye yeh har arrow ko khud par hi bhejta hai. Dekho Identity and Inverse Matrices.
Zero divisors. $N = (0010)$ ke saath, $N^{2} = (0000)$ even though $N \neq = 0$ . Geometrically $N$ puri plane ko ek line par, phir origin par squash karta hai — do squashes sab kuch annihilate kar dete hain. Toh $A B = 0$ se yeh force nahi hota ki $A = 0$ ya $B = 0$ .

YEH COLLECT KYUN KARO. Real problems mein non-square, identity, aur singular matrices aate hain; row·column rule sab ko same tarike se handle karta hai, lekin shapes aur surprises ek baar dekhne chahiye.

PICTURE. Panels: shape-locking (inner dims cancel), $I$ flag ko aise chhod raha hai, aur $N$ plane ko ek line par collapse kar raha hai.

Step 7 — Associativity free hai (grouping kabhi matter nahi karta)

KYA. Teen machines ke liye, $(A B) C = A (B C)$ .

KYUN, visually. Teen boxes $C \to B \to A$ se flow karta ek dot ka ek hi destiny hai. Chahe tum pehle last do boxes fuse karo ( $A B$ ) ya pehle do ( $B C$ ), pipeline identical hai. Kisi bhi vector par apply karo: $((A B) C) x = (A B) (C x) = A (B (C x)) = A ((B C) x) = (A (B C)) x .$ Har equality bas "ek ek box peel off karo" hai. Sab $x$ ke liye true ⟹ matrices match karti hain.

KYUN algebraically. Dono bracketings ek hi triple sum $\sum_{k} \sum_{j} A_{ik} B_{k j} C_{j ℓ}$ mein collapse ho jaati hain — aur finite sums reorder ho sakte hain (Step 3 ka trick phir se). Toh " $A B C$ " ko kisi bracket ki zaroorat nahi.

Step 5 se contrast karo: grouping free hai, lekin order nahi. Do bahut alag feel karne wale facts, dono "matrices are composed maps" ke consequences.

PICTURE. Ek line mein teen boxes; do alag fusings draw ki gayi hain — same output arrow.

Ek-picture summary

Ek figure, poori kahaani: ek arrow $B$ mein enter karta hai, ek tilted arrow ban jaata hai, $A$ mein enter karta hai, final arrow ban jaata hai — aur fused machine $A B$ yeh ek step mein karta hai, jisme uski har entry $A$ -ki-row aur $B$ -ke-column ke milne ke roop mein draw ki gayi hai.

Recall Feynman retelling (seedhe words mein)

Ek matrix ek machine hai jo grid ke dots ko naye spots par dhakelta hai, grid ko seedha rakhte hue. Uske do columns exactly batate hain ki "right" arrow aur "up" arrow kahan land karte hain — baaki sab follow karta hai. Do machines ko ek mein jodne ke liye (" $B$ karo, phir $A$ "), main poochhta hoon: kaunsi single machine har starting arrow ke liye same landing spot deti hai? Ek arrow ko dono machines se chase karke numbers collect karne par, jawab force ho jaata hai: combined machine ke slot (row $i$ , column $j$ ) fill karne ke liye, $A$ ki row $i$ ko $B$ ke column $j$ par slide karo, matched numbers multiply karo, aur add karo — yahi dot product hai, yahi poora rule hai. Kyunki right wali machine pehle run karti hai, order swap karne par usually dots alag jagah land karte hain ( $A B \neq = B A$ ) — normal hai, weird nahi. Lekin teen machines ko regroup karne se pipeline kabhi nahi badlta, isliye $(A B) C = A (B C)$ pe sochne ki zaroorat nahi. Shapes lock honi chahiye (inner sizes kiss karke disappear hote hain), identity sab kuch aise chhod deti hai, aur kuch non-zero machines plane ko itna squash karti hain ki unhe square karne par zero milta hai.

Active recall

B

ke columns geometrically kahan se aate hain?

Woh wahan hain jahan basis arrows

\overset{e}{^}_{1}, \overset{e}{^}_{2}

B

act karne ke baad land karte hain.

(A B)_{ij} = \sum_{k} A_{ik} B_{k j}

mein kaunsa index shared (summed over) hai?

k

— yeh

A

ki row

i

ke saath aur

B

ke column

j

ke neeche run karta hai.

A (B x)

mein right-hand matrix pehle kyun act karta hai?

Woh

x

ke sabse kareeb hota hai; brackets kehte hain innermost pehle, isliye

B

A

se pehle run karta hai.

Derivation mein double sum reorder karna legal kyun hai?

Yeh ordinary numbers ke finite sums hain, aur addition ko freely rearrange kiya ja sakta hai.

Associativity "free" kyun hai?

Dono groupings maps ka same pipeline hain aur same triple sum

\sum_{k, j} A_{ik} B_{k j} C_{j ℓ}

mein collapse ho jaate hain.

Kya

A B

aur

B A

ki shapes alag ho sakti hain?

Haan — e.g.

2 \times 3

times

3 \times 2

deta hai

2 \times 2

, lekin reverse deta hai

3 \times 3

Ek aisi non-zero matrix do jiska square zero ho.

(0010)

; yeh plane ko ek line par, phir ek point par squash karta hai.

Connections

Linear Transformations — "machine moves the grid" picture jo har step ke peeche hai.
Dot Product — $A B$ ki har entry ek row·column dot product hai (Step 3–4).
Identity and Inverse Matrices — do-nothing machine $I$ aur composition ko undo karna.
Determinants — ek machine kitna area scale karti hai, aur $det (A B) = det A det B$ .
Transpose — rows↔columns flip karta hai; order inverses ki tarah reverse hota hai.
Matrix Powers and Diagonalization — ek machine repeat karna, aur kab machines commute karti hain.