4.5.31 · D5 · HinglishLinear Algebra (Full)
Question bank — Diagonalization — conditions, procedure
4.5.31 · D5· Maths › Linear Algebra (Full) › Diagonalization — conditions, procedure
Yeh vocabulary yaad rakho, kabhi blur mat karo:
- Algebraic multiplicity — kitni baar ka root hai.
- Geometric multiplicity — actually kitne independent eigenvectors supply karta hai.
- Diagonalizable matlab jahan invertible hai aur diagonal hai, jo ko independent eigenvectors rakhne pe force karta hai.
True or false — justify
A matrix with a repeated eigenvalue is never diagonalizable
False — mein repeat hota hai phir bhi do independent eigenvectors milte hain (); repetition sirf tab diagonalizability ko threaten karta hai jab ho.
Every matrix with distinct eigenvalues is diagonalizable
True — distinct eigenvalues apne eigenvectors ko linearly independent force karte hain, jo ke liye independent columns de deta hai.
If a matrix is diagonalizable then it must have distinct eigenvalues
False — converse fail hota hai; ek single repeated eigenvalue ke saath diagonalizable hai, toh "distinct" sufficient hai lekin necessary nahi.
A real matrix always has a real diagonalization
False — rotation ke eigenvalues hain, toh yeh sirf pe diagonalize hota hai, pe nahi.
Diagonalizable matrices are exactly the invertible matrices
False — dono notions independent hain; diagonal hai (isliye diagonalizable) lekin invertible nahi, aur ek shear invertible hai lekin diagonalizable nahi.
If is diagonalizable then so is
True — agar hai toh same ke saath, aur diagonal hai (dekho Matrix Powers and Exponentials).
Two similar matrices always have the same eigenvalues
True — Similar Matrices same Characteristic Polynomial share karte hain, toh unke roots (eigenvalues) coincide karte hain, chahe eigenvectors na karein.
Every symmetric real matrix is diagonalizable
True — Spectral Theorem ek real orthonormal eigenbasis guarantee karta hai, toh automatically har eigenvalue ke liye hota hai.
Scaling an eigenvector changes the diagonal matrix
False — ek eigenvector aur uska koi bhi nonzero multiple same eigenvalue share karte hain, toh sirf badalta hai jabki (aur pairing) fixed rehti hai.
A matrix and its transpose are diagonalizable together or not at all
True — aur same characteristic polynomial aur same pattern share karte hain, toh ek diagonalizable hai exactly tab jab doosra bhi ho.
If every eigenvalue of equals then is the zero matrix
False — ek nilpotent shear jaise mein sirf hai phir bhi nonzero hai (aur defective, toh diagonalizable nahi).
Spot the error
" has characteristic polynomial , so it must be defective."
Error — algebraic multiplicity akela kuch nahi kehta; aapko compute karna hoga; agar hai (jaise ) toh yeh perfectly diagonalizable hai.
"I put eigenvalues in in increasing order and stacked the eigenvectors in however I liked."
Error — ka column us eigenvalue ka eigenvector hona chahiye jo mein hai; pairing scramble karna silently tod deta hai chahe same eigen-data appear kare.
" can exceed when an eigenvalue is very repeated."
Error — inequality hamesha hoti hai; geometric multiplicity kabhi algebraic multiplicity se aage nahi ja sakti.
"Since contains eigenvectors, any set of eigenvectors makes a valid ."
Error — eigenvectors linearly independent hone chahiye taaki invertible ho; ek defective matrix ke eigenvectors span nahi karte, toh koi valid exist nahi karta.
" means equals , just written differently."
Error — aur sirf similar hain (alag bases mein same transformation via Change of Basis); woh equal sirf tab hote hain jab ho, yaani jab already diagonal tha.
"The characteristic polynomial has degree , so there are always real eigenvalues, hence eigenvectors."
Error — roots complex ya repeated ho sakte hain, aur repeated roots eigenvectors under-supply kar sakte hain, toh na "real" guaranteed hai na " independent eigenvectors."
Why questions
Why must the columns of be eigenvectors and not any convenient basis?
Kyunki ko column by column expand karne se milta hai, jo exactly eigenvalue equation hai — diagonalization forced hai, chosen nahi.
Why does the condition require independent eigenvectors rather than just eigenvectors?
Kyunki invertible hona chahiye, aur ek matrix invertible hai if and only if uske columns linearly independent hain; dependent columns ek singular dete hain aur koi nahi hota.
Why do distinct eigenvalues automatically give independence?
Alag eigenvalues ke eigenvectors alag stretch-directions mein hote hain aur ek doosre se build nahi ho sakte, toh koi bhi combination jo dependency force kare zero vector pe collapse ho jaata hai — woh independent hone hi chahiye.
Why is diagonalization so useful for computing ?
Eigenbasis mein sirf har axis ko stretch karta hai, toh aur bas har diagonal entry ko th power pe raise karna hai — ek mushkil repeated multiplication ek line ban jaati hai (dekho Matrix Powers and Exponentials).
Why does a pure shear fail to be diagonalizable?
Ek shear ek direction ko fix karta hai aur baaki sab ko twist karta hai, toh uske paas sirf ek stretch-direction hota hai; ek map ke liye sirf ek independent eigenvector se koi full eigenbasis nahi banti jiske along axes redraw ho sakein.
Why can a real matrix be diagonalizable over but not over ?
Rotations ka koi real fixed direction nahi hota, toh unke eigenvalues complex hote hain (); eigenvectors bhi complex hote hain, matlab diagonalizing basis ke andar exist hi nahi karti.
Why doesn't the order you list eigenvectors change whether is diagonalizable?
ke columns reorder karna (aur ko match karna) sirf basis relabel karta hai; set phir bhi independent hai aur span karta hai, toh diagonalizability set ki property hai, uski order ki nahi.
Edge cases
Is the zero matrix diagonalizable?
Haan — yeh already diagonal hai, ke saath algebraic multiplicity hai aur se milta hai.
Is a matrix always diagonalizable?
Haan — ek akela number trivially diagonal hai, uska akela eigenvector koi bhi nonzero scalar hai, aur kabhi fail nahi hota.
If has a zero eigenvalue, can it still be diagonalizable?
Haan — ek legitimate stretch factor hai (woh us direction ko flatten karta hai); jaise ek projection eigenvalues aur ke saath diagonal hai.
Is a diagonal matrix with a repeated entry, like , diagonalizable?
Haan — yeh already diagonal hai; repeated eigenvalue ka ek full -dimensional eigenspace hai (), toh kuch bhi defective nahi hai.
Does a matrix with for some eigenvalues but not all still diagonalize?
Nahi — theorem demand karta hai ki har eigenvalue ke liye ho; ek bhi deficient eigenvalue ek independent eigenvector chhin leta hai aur ka count tod deta hai.
When has a full set of eigenvalues but one eigenspace is deficient, what structure describes it instead?
Yeh diagonalizable nahi hai lekin phir bhi ek nearly-diagonal form ke similar hai jisme diagonal ke theek upar s hain — yeh Jordan Normal Form hai, defective matrices ka canonical ghar.
Recall One-line survival test
diagonalize hota hai har eigenvalue ke liye, jo independent eigenvectors woh deta hai () woh utne hi hone chahiye jitni baar woh root ke roop mein appear hota hai (). Distinct eigenvalues aur symmetric real matrices yeh automatically pass karte hain; shears aur pe rotations classic failures hain.