Re: [線代] 一題考古題

[Sfly (topos)]

※ 引述《silentsecret ()》之銘言：
: 若A、B為n*n實矩陣，AB=BA
: 證明A、B有一共同的eigenvector
: 請問大家了!

over C, there exists some k that is an eigenvalue of A
Let V=ker(A-kI)
AB=BA => V is stable under B.
So, as a morphism, we can consider the restriction B|V

Again, there exists an eigenvector (in V) for B|V.

Then we are done.

PS.This implies that if A has n different eigenvalues,then B is diagonalizable.

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◆ From: 76.94.119.209
※ 編輯: Sfly            來自: 76.94.119.209        (02/15 16:32)
推 jack0711    :如果A.B可對角化,但存在相同λ的話敘述也會成立
推 hjmeric     :請問這裡的stable是什麼意思?
→ jack0711    :提供名師的證明:http://ppt.cc/hczy
※ 編輯: Sfly            來自: 76.94.119.209        (02/15 16:41)
→ Sfly        :stable: B maps V into V.
推 jacky7987   :B-invariant
推 hjmeric     :謝謝!