Foundations — Pseudoinverse

4.5.42 · D1 · Maths › Linear Algebra (Full) › Pseudoinverse

Yeh page kuch bhi assume nahi karta. parent page par pseudoinverse se milne se pehle, hum har woh symbol build karte hain jo woh use karta hai — ek ek brick, har brick pichle wale par tikee hoti hai. Hum pseudoinverse ka apna symbol §6 tak likhna hold karenge, jab tak ki woh plain inverse jise yeh copy karta hai table par na aa jaaye.

1. Matrix actually hai kya* (ek machine jo arrows move karti hai)

Hum ek matrix ko uski shape se maapte hain: $A\in {\mathbb{R}}^{m\times n}$ ka matlab hai "$m$ rows, $n$ columns, saare real numbers". Ise padhein out-size $\times$ in-size ke roop mein: yeh ek $n$-number arrow khaati hai aur ek $m$-number arrow produce karti hai.

Figure s01 — machine picture. Violet arrow aapka input $x$ hai; machine $A$ ise magenta output arrow $b=Ax$ par bhejta hai; dashed orange arrow woh move dikhata hai jo $A$ ne kiya. Yeh picture yaad rakho: is page par har cheez is baare mein hai ki woh move ho sakti hai ya nahi (aur kaise) reverse.

2. Columns, aur output kahan land ho sakta hai (column space)

$Ax=b$ padhne ka yeh sabse useful tarika hai. Agar $A$ ke columns $c_1,c_2,\dots ,c_n$ hain (har column khud ek arrow hai), toh

Yeh kya keh raha hai: output $b$ columns ka ek weighted sum hai, aur $x$ mein numbers weights hain. Toh $A$ sirf wahi arrows produce kar sakta hai jo aap apne columns mix karke bana sako.

Figure s02 — reachable sheet. Magenta aur violet arrows $A$ ke do columns hain. Pale orange cloud woh har arrow hai jo aap unhe mix karke bana sakte ho: woh cloud hai hi column space. Akela navy dot jisme $b$ likha hai sheet ke bahar hai — columns ka koi bhi mix use reach nahi karta, toh $Ax=b$ ka koi exact solution nahi hai.

3. Ek arrow ki length, aur norm symbol $\Vert \cdot \Vert$

"Closest" kehne ke liye hume distance measure karni hogi. Ek arrow $v=\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)$ ke liye:

Symbol $\Vert v{\Vert }^2$ (length squared) square root drop karta hai: $\Vert v{\Vert }^2=v_1^2+v_2^2$. Hum squared version prefer karte hain kyunki iska koi ugly root nahi hai, aur "smallest length" aur "smallest length-squared" ek hi jagah hote hain.

4. Dot product aur right angles ($v^{\top }w$)

Ek row $v^{\top }$ ko ek column $w$ ke saath stack karna matching entries ko multiply karta hai aur unhe add karta hai — yeh dot product hai:

Figure s03 — right-angle test. Magenta arrow $v$ aur violet arrow $w$ ek corner par milte hain jo chhote orange square se marked hai (standard "right angle" symbol). Unka dot product exactly $0$ hai. Picture padhein: dot product zero $\iff$ perpendicular.

5. Projection: sheet par shadow

Inhe saath rakhne par aapko star player milta hai.

Parent page par aap padhenge "$A{A}^{+}$ column space par orthogonal projection hai" (symbol $A^{+}$ §6 mein neeche naam diya gaya hai). Iska matlab: $b$ ko $A$ se aur phir fair-undo machine se feed karna $b$ khud return nahi karta; yeh reachable sheet par $b$ ka shadow return karta hai. Poori picture ke liye Orthogonal Projection dekho.

6. Plain inverse $A^{-1}$, aur uska fair sibling $A^{+}$

$A^{-1}$ do situations mein exist karna fail karta hai:

7. Nullspace aur row space — ek hi output de sakte hain bahut saare inputs kyun

Jab machine "squishes" karti hai, kai alag input arrows ek hi output par collapse ho jaate hain. Jinhe zero mein squish kiya jaata hai unka naam hai.

Yeh bahut saare solutions kyun create karta hai. Maano $A{x}_0=b$ aapka problem solve karta hai. Koi bhi nullspace arrow $n$ lo (toh $An=0$). Toh $A(x_0+n)=A{x}_0+An=b+0=b,$ toh $x_0+n$ bhi solve karta hai. Solutions ki poori family, har nullspace arrow ke liye ek! Is case mein pseudoinverse ka kaam us family ka sabse short wala member chuna hai — §3 ka minimum-norm idea dekho.

Figure s04 — squish aur shortest choice. Machine poore plane ko magenta line (uska column space) par flatten kar deti hai; violet dashed line nullspace hai (arrows squished to zero). Saare navy dots woh inputs hain jo ek hi output dete hain; filled orange dot sabse short wala hai — yeh nullspace ke perpendicular baitha hai, exactly woh minimum-norm answer jo $A^{+}$ return karta hai.

8. Rank — machine kitna flatten karti hai

Rank teen cases label karta hai jo parent solve karta hai:

• Full column rank (rank $=n$, tall & thin): columns sab independent — andar jaate waqt kuch bhi waste nahi, aur nullspace bas zero arrow hai.
• Full row rank (rank $=m$, short & fat): output sab kuch reach kar sakta hai — bahut saare inputs yeh karte hain, toh ek nontrivial nullspace minimum-norm choice force karta hai.
• Rank-deficient: kuch directions zero par collapse hote hain; sirf Singular Value Decomposition bachti hai.

9. Singular values ${\sigma }_i$ — stretch factors

Ek ${\sigma }_i=0$ ka matlab hai ellipse ki ek direction ek point par squash ho gayi — information destroy ho gayi, aur woh squashed direction exactly §7 ki ek nullspace direction hai. ${\sigma }_i$ se stretch ko ${\sigma }_i$ se divide karke undo kar sakte ho… jab tak ${\sigma }_i=0$ na ho, kyunki aap ek point ko wapas ek line mein un-squash nahi kar sakte (koi $1/0$ nahi hota). Yahi poora reason hai ki SVD pseudoinverse "zeros ko zeros rakhta hai". ${\sigma }_i^2$ $A^{\top }A$ ke eigenvalues hain — Eigenvalues and Eigenvectors dekho.

Prerequisite map

Equipment checklist

Neeche khud ko test karo. Har item prompt ::: answer ke roop mein likha hai — sirf left side padhein, apna jawab zor se bolo, phir right side se check karo.

Connections

• Pseudoinverse — woh topic jiske liye yeh foundations build ho rahe hain.
• Singular Value Decomposition — singular values kahan se aate hain.
• Orthogonal Projection — "closest shadow" precisely banaya gaya.
• Matrix Inverse — woh perfect undo jo $A^{+}$ generalize karta hai.
• Four Fundamental Subspaces — column space, row space, nullspace aur left nullspace.
• Eigenvalues and Eigenvectors — ${\sigma }_i^2$ $A^{\top }A$ ke eigenvalues hain.
• Least Squares Regression — projection idea ka payoff.