Foundations — QR decomposition

4.5.36 · D1 · Maths › Linear Algebra (Full) › QR decomposition

Yeh page assume karta hai ke tumne kuch bhi nahi dekha. Hum har woh symbol build karenge jo parent note use karta hai, ek ek karke, har ek pichle pe tika hua. Upar se neeche padho.

0. Vector asal mein hota kya hai?

Picture: origin se ek seedha arrow. Bas itna hi.

Topic ko yeh kyun chahiye. Parent note likhta hai ${\mathbf{a}}_1,{\mathbf{a}}_2,\dots$ — woh bas woh arrows hain jinse hum shuru karte hain. QR jo bhi karta hai woh arrows ko rearrange karta hai, toh "arrow = numbers ki list" line one hai.

1. Matrix as column-arrows ka stack

Picture: $n$ alag alag arrows, har ek $m$ numbers se describe kiya gaya. Parent ka example $A$ ka $m=3$ hai (arrows 3D space mein) aur $n=2$ hai (do arrows).

Topic ko yeh kyun chahiye. QR ek statement hai columns ke baare mein. $A$ ko column-arrows ke stack ke roop mein padhna woh poori mental switch hai jo baaki sab ko visual banati hai.

2. Arrow ki length: norm $\Vert \mathbf{a}\Vert$

Topic ko yeh kyun chahiye. Parent likhta hai $r_{kk}=\Vert {\mathbf{u}}_k\Vert$ aur "normalize" — dono length ke baare mein hain. Arrow ko length 1 tak shrink karna tab tak possible nahi jab tak uski length pata na ho.

3. Dot product ${\mathbf{q}}^{\top }\mathbf{a}$ — yahan sabse important tool

Topic ko yeh kyun chahiye. Parent note mein har $r_{ik}={\mathbf{q}}_i^{\top }{\mathbf{a}}_k$ ek dot product hai. Aur "orthonormal columns," "$Q^{\top }Q=I$," aur "perpendicular" sab dot products ke zero hone par khatam hote hain. Yeh seekh lo aur QR ka aadha hissa tumhara hai.

4. Projection — ek arrow ka doosre par "shadow"

Aao actually check karte hain woh last claim, kyunki yeh woh trick hai jis par poora parent page chalta hai. $\mathbf{q}$ aur $\mathbf{u}$ ka dot product lo: ${\mathbf{q}}^{\top }\mathbf{u}={\mathbf{q}}^{\top }\mathbf{a}-({\mathbf{q}}^{\top }\mathbf{a})\underset{=1}{\underbrace{{\mathbf{q}}^{\top }\mathbf{q}}}={\mathbf{q}}^{\top }\mathbf{a}-{\mathbf{q}}^{\top }\mathbf{a}=0.$ Humne kya kiya: shadow subtract kiya. Kyun: $\mathbf{q}$ ke saath koi bhi overlap khatam karne ke liye. Kaisa dikhta hai: bacha hua arrow $\mathbf{u}$ $\mathbf{q}$ ke saath perfect right angle par khada hai. Yeh exactly parent ka "jo pehle se explain ho chuka hai use hatao" wala step hai.

Topic ko yeh kyun chahiye. Parent ka ${\mathbf{u}}_2={\mathbf{a}}_2-({\mathbf{q}}_1^{\top }{\mathbf{a}}_2){\mathbf{q}}_1$ hai hi shadow-subtract karna. Projection Gram–Schmidt ka engine hai, jo QR ka engine hai — dekho Orthonormal bases and projections.

5. Orthonormal — perpendicular aur length 1

Picture: arrows jaise ek kamre ka saaf kona — upar, daayein, aage — har ek length 1 ka, sab right angles par. Yahi woh "clean coordinate frame" hai jiska parent baar baar zikr karta hai.

Topic ko yeh kyun chahiye. $A=QR$ mein $Q$ ke columns orthonormal hain. Yahi cheez $Q$ ko lengths aur angles preserve karne wala (rotation/reflection) banati hai.

6. Identity matrix $I$ aur $Q^{\top }Q=I$ ka matlab

Topic ko yeh kyun chahiye. Yeh parent ki $Q$ ki definition hai. Ab tum ise padh sakte ho instead of ratt marne ke.

7. Upper-triangular — diagonal ke neeche zeros

Topic ko yeh kyun chahiye. $A=QR$ mein $R$ upper-triangular hai. Parent kyun prove karta hai (har ${\mathbf{a}}_k$ sirf pehle ke $\mathbf{q}$'s use karta hai); yahan tumhe bas shape pehchaanni hai. Sibling LU decomposition bhi triangular matrices use karta hai.

8. Linear independence — "koi bhi arrow redundant nahi"

Picture: teen arrows jo genuinely teen alag directions mein pahunchte hain, sab ek line ya ek plane par nahi dabbe hue.

Topic ko yeh kyun chahiye. Yeh parent ki definition mein ek requirement hai ($A$ ke columns linearly independent hain). Yahan Gram-Schmidt process bina iske break ho jaata.

Prerequisite map

Ise aise padho: arrows sab kuch feed karte hain; dot product projection aur orthonormality ko feed karta hai; woh dono $Q$ banate hain; bacha hua lengths $R$ banata hai; saath mein woh $A=QR$ hain.

Equipment checklist

Khud test karo — har line question ::: answer format mein hai. Daayein side chhupa lo.

Jab upar ki har line easy lage, QR decomposition par wapas jaao aur derivation plain English jaisi padhegi.