Re: [線代] AB=O

[kusoayan (瑋哥)]

※ 引述《mqazz1 (無法顯示)》之銘言：
: Let A, B denote two n*n matrices satisfying AB=O.
: Then in the following, pick up the correct statements.
: (a) BA=O
False,
let  A=[ 1 0 ]   B=[ 0 0 ]
       [ 0 0 ]     [ 1 0 ]
then AB = 0, but BA != 0

: (b) all eigenvalues of BA are 0
: (c) (BA)^2 = O
True,
(BA)^2 = BABA = B(AB)A = 0

: (d) A=O or B=O
False, Similary to (a)

: (e) rank(A) + rank(B) = n
False,
pick A=B=0

: 謝謝

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推 bineapple   :b也是true 從c可得知
推 bennygameii :不好意思 我可以問一下為什麼b是true嘛??
推 jacky7987   :因為BA的特徵多項式跟AB一樣
推 PaulErdos   :如果v是BA的eigen vector, 則(BA)^2 v ＝ BA(λv)
→ PaulErdos   :＝λ^2 v ＝ 0 , 因v非零所以λ被迫要是0
推 G41271      :原來如此!!