Question bank — Eigenvalue computation — power method, inverse iteration

False — yeh magnitude $|\lambda |$ mein largest ki taraf converge karta hai. Ek matrix jiske eigenvalues $-9$ aur $+4$ hain, uska dominant $\lambda =-9$ hai, $4$ nahi.

False — ratio $|{\lambda }_2/{\lambda }_1|=1$ kabhi decay nahi karta, isliye woh dono components bane rehte hain aur iterate $v_1,v_2$ plane ke andar oscillate ya rotate karta rehta hai, settle nahi hota.

False — ise sirf ek dominant eigenvalue aur ek full eigenbasis chahiye. Symmetry ek bonus hai: yeh Rayleigh quotient ko $O({\text{error}}^2)$ accurate banata hai, $O(\text{error})$ ki jagah nahi.

False — power method ki rate $|{\lambda }_2/{\lambda }_1|$ hai jabki inverse iteration ki rate $|{\lambda }_n/{\lambda }_{n-1}|$ hai; yeh different eigenvalue gaps par depend karte hain aur usually bahut alag hote hain.

False aur dangerous — tab $A-\sigma I$ exactly singular ho jaata hai, isliye system $(A-\sigma I)y=x$ ka koi unique solution nahi hota. Practice mein near-singularity theek hai (helpful bhi); exact singularity solve ko tod deti hai.

False — normalization sirf length ko rescale karta hai; direction untouched rehti hai. Yeh sirf isliye hota hai taaki vector overflow (${\lambda }_1^k$) ya underflow na kare.

False — yeh current vector ke liye ek best-fit eigenvalue hai, isliye yeh har step par ek usable (improving) estimate deta hai, convergence se bahut pehle.

True — $Av=\lambda v$ se, $A^{-1}/\lambda$ se multiply karke $A^{-1}v=\frac{1}{\lambda }v$ milta hai; same $v$, reciprocal eigenvalue. Dekho Spectral Decomposition.

False — $(A-\sigma I)v=(\lambda -\sigma )v$ same $v$ rakhta hai; sirf eigenvalues $\sigma$ se slide karte hain. Yahi invariance reason hai ki shift-and-invert ek chosen eigenvector ko target kar sakta hai.

Result sahi hai lekin method galat hai: explicitly invert karna $O(n^3)$ hai, numerically unstable hai, aur wasteful hai. Iske bajaye har step mein $A{y}_{k+1}=x_k$ solve karo, ek LU Decomposition reuse karke — "solve karo, invert mat karo."

$A^k$ ka fast badhna direction ka convergence nahi hai. Direction mein error $|{\lambda }_2/{\lambda }_1{|}^k$ ki tarah decay karta hai — har step mein ek constant ratio, matlab linear convergence, quadratic nahi.

$c_1$ ek specific seed ka $v_1$-component hai; ek alag seed jo $\text{span}(v_2,\dots ,v_n)$ mein padi ho, uska $c_1=0$ hoga. Tumhe har seed check karni chahiye (random seeds almost surely kaam karti hain).

Rate actually ratio $\left|\frac{{\lambda }_j-\sigma }{{\lambda }_m-\sigma }\right|$ hai jahan ${\lambda }_m$ doosra sabse kareeb eigenvalue hai $\sigma$ ke. Speed ko sirf ratio control karta hai (numerator vs. next eigenvalue tak distance), raw distance nahi.

$n\ge 5$ ke liye koi closed-form radical formula nahi hai (Characteristic Polynomial), lekin eigenvalues phir bhi iterative methods jaise in methods se ya QR Algorithm se kisi bhi precision tak computable hain.

Plain power method ${\lambda }_1$ return karta rehta hai — dominant direction phir se dominate kar leti hai. ${\lambda }_2$ tak pahunchne ke liye tumhe deflate karna hoga ($v_1$ component hatao) ya shift/inverse strategy use karni hogi.

Yeh sirf tab valid hai jab $x$ already normalized ho ($x^{\top }x=1$). Generally $\mu =\frac{x^{\top }Ax}{x^{\top }x}$; unnormalized vector par denominator drop karne se tumhara eigenvalue estimate rescale ho jaata hai.

Largest term factor out karne se ratios $({\lambda }_i/{\lambda }_1{)}^k$ milte hain jo sab $<1$ hain aur vanish ho jaate hain; kuch chhota factor out karne par ek term $\ge 1$ reh jaati jo blow up karti, convergence hide ho jaata.

$A^{-1}$ ke eigenvalues $1/{\lambda }_i$ hote hain, isliye $A$ ka sabse chhota $|\lambda |$, largest $|1/\lambda |$ ban jaata hai — woh wala jo power method naturally amplify karta hai. Dekho Convergence Rates.

${\lambda }_j-\sigma$ tiny ho jaata hai, isliye $\frac{1}{{\lambda }_j-\sigma }$ enormous ho jaata hai aur har doosre reciprocal ko dwarf kar deta hai — next term ka ratio zero ki taraf collapse karta hai, isliye ek ya do steps kaafi ho sakte hain.

Rayleigh quotient current vector ki sab directions par least-squares best eigenvalue hai, isliye yeh not-yet-decayed error components ko average out karta hai; ek single component jis non-dominant piece se corrupt hota hai woh wahan hoti hai.

Matrix $A-\sigma I$ har iteration mein same hota hai; sirf right-hand side $x_k$ change hota hai. Ek $O(n^3)$ factorization plus saste $O(n^2)$ forward/back-substitutions har step mein repeated full solves se better hai.

Non-dominant error $|{\lambda }_2/{\lambda }_1{|}^k$ ki tarah decay karta hai; agar woh ratio $\approx 0.99$ hai, toh har step sirf $1\%$ error hatata hai, isliye converge karne mein saikdon iterations lagte hain.

Exact arithmetic mein $A^k{x}_0={\lambda }_2^k{v}_2$ hamesha — tum galat eigenvector par converge karte ho. Floating point mein, rounding usually ek tiny $v_1$ component wapas inject kar deta hai jo eventually dominate karta hai.

$A$ singular hai, isliye $A^{-1}$ (ya solve $Ay=x$) exist nahi karta — inverse iteration start nahi ho sakta. Shifted inverse iteration ke liye yeh theek hai jab tak $A-\sigma I$ nonsingular ho.

Tab $|{\lambda }_1|=|{\lambda }_2|$ hota hai, do complex conjugates magnitude mein tied hote hain, isliye real power method ek single real eigenvector pick nahi kar sakta aur instead oscillate karta hai — yeh signal hai QR Algorithm switch karne ka.

Do sabse kareeb eigenvalues equidistant hain, isliye $|{\lambda }_j-\sigma |=|{\lambda }_m-\sigma |$ aur convergence ratio $1$ ho jaata hai — shifted inverse iteration stall kar jaata hai; $\sigma$ ko us wale ki taraf nudge karo jo tumhe actually chahiye.

Symmetric $A$ ke liye Rayleigh quotient error $O({\text{eigenvector error}}^2)$ hota hai, isliye ${\mu }_k$ vector se kahin zyada accurate hota hai aur exponent mein roughly do guna fast converge karta hai.

Iterate phir bhi $v_1$ direction ke saath align ho jaata hai (sign tak); $-v_1$ utna hi valid eigenvector hai jitna $v_1$, aur Rayleigh quotient sign regardless same ${\lambda }_1$ return karta hai.