[理工] [線代] 100交大資聯

[mickeyha (M*schief)]

黃天王題庫班5-126

10.Suppose A(3X3) = xy^T is a rank-1 matrix.
   Which of the following statements are true?

(a)The eigenvector matrix S(i.e.,AS=VS) must be invertible.
(b)A must have one non-zero eigenvalue.
(c)When A isfactorized by singular value decomposition(SVD)
   into U希格馬V^T, the only non-zero singular value is ||x||||y||.
(d)When A is factorized by SVD into U希格馬V^T,
   U must include x/||x|| as ione of its column vectors.

(抱歉...有些符號找不到該怎麼打>_<)

答案為(c)(d)

這題我卡了很久
應該是觀念不夠健全...
煩請板上高手指點了!!!

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Why Not ：-P

http://whynot-p.blogspot.com/

是的我是江米奇別再在宿舍走廊上問我了>////<

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推 pikachu123:他的eigenvalue為y^tx,0,0
→ pikachu123:(a)gm(0)=2,but y^tx=0時am(0)=3 A會不可對角化
→ pikachu123:他的eigenvetor matrix不一定可逆 你找jordan form就是
→ pikachu123:(b)同(a)之講法
推 pikachu123:(c)A^tA=(xy^t)^t(xy^t)=(x^tx)(y^ty)=||x||||y||
→ pikachu123:(c)打錯那個A^tA=||x||yy^t 他的非零eigenvlue
→ pikachu123:就把yy^t反過來 就是(x^tx)(y^ty)=||x||||y||
推 pikachu123:(d)觀察一下 y為A相對||x||||y||之eigenvector
→ pikachu123:               A^tA(又打錯=.=)
→ pikachu123:則v1=y/||y|| u1=(1/||x||y||)Av1
→ pikachu123:u1=(1/||x||||y||)xy^t(y/||y||)
→ pikachu123:=(1/||x||(||y||^2))xy^ty=(1/||x||(||y||^2))x||y||^2
→ pikachu123:=x/||x||
→ pikachu123:(d)就是考你會不會SVD
→ pikachu123:這題我看是100交大線代最難的 SVD+rank1...
推 pikachu123:(d)其實有速解 xy^t=||x||||y||(x/||x||)(y^t/||y||)
→ pikachu123:理由在你的黃天王講義交大的下一題 不嚴謹就是...
推 pikachu123:A之SVD 可以寫成A=σ1u1v1^t+σ2u2v2^t+...+σrurvr^t
→ pikachu123:類似光譜分解
推 Jerrynet:這題x和y是vector嗎？
→ gskman:這題是頗難的
→ sneak: (d)其實有速解 xy https://daxiv.com