Foundations — Quadratic forms — positive definite, negative definite, indefinite

4.5.39 · D1 · Maths › Linear Algebra (Full) › Quadratic forms — positive definite, negative definite, inde

Parent note Quadratic forms — positive definite, negative definite, indefinite padhne se pehle, tumhe usme use hone waale har symbol ko khud se samajhna hoga. Yeh page har ek concept ko ek-ek karke, ek-doosre pe depend karne ke order mein, zero se build karta hai.

1. Vector $\mathbf{x}$ — numbers se bana ek arrow

Picture: 2D mein, $\mathbf{x}=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$ origin se point $(x_1,x_2)$ tak ek arrow hai. Figure dekho: red arrow hi vector hai; uske do components batate hain ki woh kitna right aur kitna upar jaata hai.

Topic ko yeh kyun chahiye: machine $Q$ ek vector khata hai aur ek number deta hai. Neeche sab kuch usi arrow ke saath kya hota hai, usi ke baare mein hai.

2. ${\mathbb{R}}^n$ — woh space jisme arrow rehta hai

Picture: ${\mathbb{R}}^2$ flat plane hai (ek sheet of paper par har point). ${\mathbb{R}}^3$ poora space hai. $\mathbf{x}\in {\mathbb{R}}^n$ likhna (padho "$\mathbf{x}$ in ${\mathbb{R}}^n$") bas yeh kehta hai ki "$\mathbf{x}$ unhi arrows mein se ek hai".

Topic ko yeh kyun chahiye: definiteness ka sawaal hai "saare $\mathbf{x}\in {\mathbb{R}}^n$ ke liye, $Q$ ka sign kya hai?" — humein arrows ke poore space pe jaana hoga, sirf ek par nahi.

3. Matrix $A$ — numbers ki ek grid

Picture: $A$ ko ek transformation samjho — ek rule jo kisi bhi arrow ko ek nayi jagah le jaata hai. Yeh space ko stretch, squash, aur rotate karta hai. Quadratic form mein yeh machine ki recipe store karta hai.

Topic ko yeh kyun chahiye: poora quadratic form $A$ mein packaged hai. Diagonal entries squared-term coefficients carry karti hain; off-diagonals cross terms carry karti hain.

4. Transpose $A^{\top }$ aur ${\mathbf{x}}^{\top }$ — diagonal ke along flip karo

Picture: diagonal ko fixed rakho aur baaki sab ko uske through reflect karo — jaise paper ko top-left-to-bottom-right crease ke along fold karna. Figure ek matrix aur uski mirror image dikhata hai.

Topic ko yeh kyun chahiye: form ${\mathbf{x}}^{\top }A\mathbf{x}$ likhaa jaata hai — aage wala ${\mathbf{x}}^{\top }$ ek row hai, aur "symmetric" ($A^{\top }=A$) woh key property hai jo parent demand karta hai.

5. Symmetric — mirror $A$ ko unchanged chhod deta hai

Picture: ek symmetric matrix diagonal ke across Rorschach blot jaisi dikhti hai. $\left(\begin{array}{cc}3& 2\\ 2& 1\end{array}\right)$ mein do $2$'s mirror twins hain.

Topic ko yeh kyun chahiye: sirf ek matrix ka symmetric part hi ${\mathbf{x}}^{\top }A\mathbf{x}$ ko affect karta hai, isliye hum hamesha $A$ symmetric choose karte hain. Symmetric matrices ki ek khoobsoorat guarantee bhi hoti hai (real eigenvalues, perpendicular eigenvectors) — dekho Symmetric Matrices aur Spectral Theorem.

6. Matrix–vector multiplication aur ${\mathbf{x}}^{\top }A\mathbf{x}$

$2\times 2$ ke liye yeh cleanly expand hota hai. $A=\left(\begin{array}{cc}a& b\\ b& c\end{array}\right)$ ke saath: ${\mathbf{x}}^{\top }A\mathbf{x}=a{x}_1^2+2b{x}_1{x}_2+c{x}_2^2.$

Topic ko yeh kyun chahiye: yahi quadratic form hai. Parent mein har result isi ek number ke sign ke baare mein ek statement hai.

7. Scalar — ek akela number, aur yeh apne transpose ke barabar kyun hota hai

Topic ko yeh kyun chahiye: parent "ek scalar apne transpose ke barabar hota hai" ka use ek matrix ka antisymmetric part khatam karne ke liye karta hai. Woh trick tabhi kaam karti hai kyunki ${\mathbf{x}}^{\top }A\mathbf{x}$ ek scalar hai.

8. Degree-2 / homogeneous — har term exactly do coordinates ka product hai

Picture: agar tum input arrow double karo, output number chaar guna ho jaata hai ($Q(2\mathbf{x})=4Q(\mathbf{x})$) — pure degree-2 ki pehchaan. Neeche figure parabola-jaisi scaling dikhata hai.

Topic ko yeh kyun chahiye: kyunki koi linear ya constant terms nahi hain, $Q(0)=0$ hamesha hota hai. Isliye poora classification ka sawaal sirf $\mathbf{x}\ne 0$ ke liye hi mayne rakhta hai — yahi reason hai ki har definition kehti hai "for all $\mathbf{x}\ne 0$".

9. Eigenvalues $\lambda$ aur eigenvectors — woh directions jinhein $A$ sirf stretch karta hai

Picture: zyaadatar arrows $A$ ke andar ek nayi direction mein swing karte hain; ek eigenvector apni line par rehta hai. Figure mein red arrow same line par land karta hai, bas $\lambda$ se rescaled.

Topic ko yeh kyun chahiye: master result yeh hai ki eigenvector coordinates mein machine ban jaati hai $Q={\lambda }_1{y}_1^2+{\lambda }_2{y}_2^2+\dots$ — pure squares. Phir ${\lambda }_i$ ke signs sab kuch decide karte hain. Full details Eigenvalues and Eigenvectors mein hain.

10. $\Lambda$, $\text{diag}$, aur orthogonal $Q$

Topic ko yeh kyun chahiye: Spectral Theorem $A=Q\Lambda Q^{\top }$ likhta hai — "nice axes ki taraf rotate karo, $\lambda$'s se stretch karo, wapas rotate karo". $\mathbf{y}=Q^{\top }\mathbf{x}$ set karne se tangled form clean $\sum {\lambda }_i{y}_i^2$ mein badal jaati hai.

11. Determinant $\det$ aur leading principal minors $D_k$

Topic ko yeh kyun chahiye: Sylvester's criterion sirf inhi $D_k$'s se definiteness test karta hai — koi eigenvalues nahi chahiye. Dekho Determinants.

Prerequisite map

Left side par sab raw material hai; arrows dikhate hain ki har idea kaisi definiteness ke sawaal ko feed karta hai, jo bilkul right par hai.

Equipment checklist

Self-test — kya tum parent note kholne se pehle har ek jawab de sakte ho?