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※ 引述《mqazz1 (無法顯示)》之銘言: : 3*3 matrix A : 4 vector: a, b, c, d : Aa=b : Ab=c : Ac=d : 請問這樣為什麼d會是a,b,c的linear combination呢? : 該怎麼推呢? : 謝謝! Cayley-Hamilton is too powerful for this question. It's easy to prove that V:=<a,Aa,A^2a,A^3a,...> = <a,Aa,A^2a>. (for example, let k be the smallest integer such that A^k(a) is in <a,Aa,..,A^(k-1)a> . Then V=<a,Aa,..,A^(k-1)a> and dimV=k. Clearly dimV<=3, so k<=3 and V=<a,Aa,A^2a>.) so A^3a = d is in <a,Aa,A^2a>=<a,b,c>. -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 76.94.119.209 ※ 編輯: Sfly 來自: 76.94.119.209 (01/02 19:45)
Lpspace :This is the same. Your claim is not so easy. XD 01/02 20:55
Sfly :well...C-H is definitly harder than my claim 01/02 23:49
herstein :yes...二樓說的對 01/03 00:22