作者JohnMash (Paul)
看板Math
标题Re: [线代] 几个命题的真伪
时间Mon Feb 14 07:27:45 2011
※ 引述《Madroach (∞)》之铭言：
: 写题目的时候碰到几个不确定的叙述
: 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]
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 112.104.89.13
1F：推 Madroach :我懂了，谢谢前辈! 02/14 10:07