Re: [線代] 幾個命題的真偽
※ 引述《Madroach (∞)》之銘言：
: 寫題目的時候碰到幾個不確定的敘述
: 1)A and B are n*n matrices, AB = O, then all eigenvalues of BA are 0.
Suppose λ an eigenvalue, and x≠0 an eigenvector of B.A
then
B.A.x=λx........(1)
then
A.B.A.x=0=λA.x.......(2)
By (2), λ=0 or A.x=0
if λ=0, done.
if A.x=0, then by (1), λ=0, done.
: 2)A is a n*n matrix over R s.t A^2=-I_n, then
: ( i ) n must be even
0≦det(A)^2=det(-I_n)=(-1)^n
done
: (ii ) tr(A)≠0
[0 1][0 1] = [-1 0]
[-1 0][-1 0] [0 -1]
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02/14 10:07, , 1F
02/14 10:07, 1F
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