Foundations — Symmetric matrices — spectral theorem (real eigenvalues, orthogonal eigenvectors)

4.5.38 · D1 · Maths › Linear Algebra (Full) › Symmetric matrices — spectral theorem (real eigenvalues, ort

Is page par kuch bhi assume nahi kiya gaya. Spectral Theorem ki ek bhi line padhne se pehle, aapko symbols ka ek chhota toolkit chahiye. Hum har ek ko ek picture se build karte hain, us order mein jisme ek doosre par depend karte hain.

1. Ek vector — ek arrow jo ek address ke saath aata hai

Ek arrow ki picture banao jo origin (point $(0,0)$) se shuru ho aur address $(x_1,x_2)$ par khatam ho. Numbers hi arrow hain.

Topic ko yeh kyun chahiye: poora theorem is baare mein hai ki ek matrix arrows ko kaise move karta hai, aur kaunse special arrows apni direction rakhte hain.

2. Ek matrix — ek machine jo arrows khaati hai

$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right),A\mathbf{x}=\left(\begin{array}{c}a{x}_1+b{x}_2\\ c{x}_1+d{x}_2\end{array}\right).$

Rule ko is tarah padho: $A$ ki har row ek recipe hai. Row 1 kehti hai "$a$ pehle coordinate ka aur $b$ doosre ka milake naya pehla coordinate deta hai." $n$ dimensions mein same recipe har row mein $n$ terms ke upar chalti hai.

Topic ko yeh kyun chahiye: $A$ show ka star hai. Theorem exactly batata hai ki symmetric $A$ space ke saath kya karta hai.

3. Ek vector ko scale karna — $\lambda \mathbf{x}$ aur $c\mathbf{x}$

Picture: $c\mathbf{x}$ same arrow hai, jiski length $c$ se multiply ho gayi. Agar $c=2$ toh double ho jaata hai; agar $c=-1$ toh ulti direction mein flip ho jaata hai; agar $c=0$ toh origin par crush ho jaata hai.

Topic ko yeh kyun chahiye: eigen-equation $A\mathbf{x}=\lambda \mathbf{x}$ ek scaling statement hai, aur $A-\lambda I$ ek scaled identity subtract karta hai — dono ko padhne ke liye yeh jaanna zaroori hai ki "vector/matrix ko scale karna" ka matlab kya hai.

4. Transpose $A^{\top }$ — diagonal ke across flip karna

${\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}^{\top }=\left(\begin{array}{cc}a& c\\ b& d\end{array}\right).$

$b$ aur $c$ apni jagah badal lete hain; $a$ aur $d$ mirror line par baithe hain aur wahi rehte hain. ($n$ dimensions mein same swap har off-diagonal pair ke liye hota hai.)

Topic ko yeh kyun chahiye: "symmetric" ki poori definition transpose ke saath likhi gayi hai.

5. Symmetric matrix — $A=A^{\top }$

$\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)={\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)}^{\top }✓$

Topic ko yeh kyun chahiye: yeh single equation $A=A^{\top }$ woh hypothesis hai jo real eigenvalues aur perpendicular axes ko unlock karti hai.

6. Identity $I$ aur zero vector $0$

Zero vector $0=\left(\begin{array}{c}0\\ 0\end{array}\right)$ zero length ka arrow hai, origin par baitha hoa. Jab hum eigenvectors dhundhte hain toh hum hamesha $\mathbf{x}\ne 0$ maangtte hain, kyunki zero arrow sab kuch trivially satisfy karta hai aur kuch nahi batata.

Topic ko yeh kyun chahiye: neeche $A-\lambda I$ phrase ek scaled identity $\lambda I$ subtract karta hai (§3 se scalar multiplication), aur eigenvector hunt $0$ ko exclude karta hai.

7. Eigenvalue $\lambda$ aur eigenvector — woh arrows jo apni direction rakhte hain

• Agar $\lambda =2$, toh arrow ki length double ho jaati hai, same direction.
• Agar $\lambda =1$, toh bilkul move nahi karta.
• Agar $\lambda =-1$, toh peeche ki taraf flip ho jaata hai (same line, opposite way).
• Agar $\lambda =0$, toh arrow origin par crush ho jaata hai.

Poore treatment ke liye dekho Eigenvalues and Eigenvectors. Expression $A-\lambda I$ inhe dhundhta hai: $\det (A-\lambda I)=0$ solve karne se $\lambda$'s milte hain.

Topic ko yeh kyun chahiye: theorem ek symmetric matrix ke eigenvalues aur eigenvectors ke baare mein ek statement hai.

8. Dot product ${\mathbf{u}}^{\top }\mathbf{v}$ — perpendicularity detector

Magical fact: ${\mathbf{u}}^{\top }\mathbf{v}=0$ exactly tab jab do arrows $90^{\circ }$ par milte hain — woh orthogonal hain (yani "perpendicular" ka fancy word).

Topic ko yeh kyun chahiye: "distinct eigenvalues ke eigenvectors orthogonal hote hain" ek statement hai ki dot product zero hai. Is tool ke bina theorem ki doosri promise padhna impossible hai.

9. Matrix inverse $A^{-1}$ — "undo" machine

Picture: agar $A$ rubber sheet ko stretch aur rotate karta hai, toh $A^{-1}$ use un-stretch aur un-rotate karke original grid par wapas le aata hai. (Har matrix ka inverse nahi hota — jo space ko flat crush karta hai use undo nahi kiya ja sakta — lekin neeche jo matrices hum build karte hain unka hamesha hota hai.)

Topic ko yeh kyun chahiye: theorem $A=Q\Lambda Q^{\top }$ likhta hai, aur jahan $Q^{-1}$ ki expect ki jaati wahan $Q^{\top }$ use karne ki wajah yeh surprising identity hai $Q^{-1}=Q^{\top }$. Yeh gift tab tak samajh nahi aata jab tak pata na ho ki $Q^{-1}$ ka matlab kya hai.

10. Orthonormal, aur matrix $Q$

"Length 1" ka matlab hai hum normalize karte hain: vector ko uski length $\sqrt{u_1^2+u_2^2}$ se divide karo (${\mathbb{R}}^n$ mein, dot product ${\mathbf{u}}^{\top }\mathbf{u}$ ka square root). Misal ke taur par $(1,1)$ ki length $\sqrt{2}$ hai, isliye iska normalized version $\frac{1}{\sqrt{2}}(1,1)$ hai.

Topic ko yeh kyun chahiye: $Q$ perpendicular skeleton directions ko unit columns ke roop mein hold karta hai; yeh woh coordinate frame hai jisme $A$ diagonal dikhta hai.

11. Diagonal matrix $\Lambda$ — pure stretching

Yeh sabse simple possible machine hai: pehle axis ko ${\lambda }_1$ se scale karo, doosre ko ${\lambda }_2$ se, khatam. Diagonal entries precisely eigenvalues hain.

Topic ko yeh kyun chahiye: theorem ka punchline $A=Q\Lambda Q^{\top }$ kehta hai "acche frame mein rotate karo ($Q^{\top }$), pure stretching karo ($\Lambda$), wapas rotate karo ($Q$)." Diagonalization isko fully unpack karta hai.

12. Determinant $\det$ — eigenvalue-finder

$2\times 2$ matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ ke liye, determinant $ad-bc$ hai.

Topic ko yeh kyun chahiye: har worked example mein eigenvalues $\det (A-\lambda I)=0$ solve karke dhundhte hain.

Yeh theorem ko kaise feed karte hain

Map ko top-to-bottom padhna is poore page ko retrace karta hai: hum real numbers aur arrows se shuru karte hain, unhe scale karna aur flip karna seekhte hain, aur theorem ke decomposition tak pahunchtein hain. Har box upar banaye gaye plain words mein ek concept name karta hai (koi naya notation nahi).

Theorem neeche baithta hai: symmetry plus perpendicular eigen-directions plus pure stretching milke $A=Q\Lambda Q^{\top }$ bante hain.

Equipment checklist

Khud test karo — right side cover karo aur zor se jawab do.