Re: [線代] 請問一題線代
※ 引述《mankato (豬頭)》之銘言：
: 請問一題線代
: A為3x2矩陣 B為2X3矩陣
: 8 2 -2
: AB=[ 2 5 4 ]
: -2 4 5
: 求BA?
: 想了很久,一直不知道如何下手,直覺跟對稱有關
: 但一直沒有頭緒
: 請大大們解惑一下
: 謝謝
char poly of AB = x(x-9)^2.
It follows that char poly of BA = (x-9)^2.
(cf. http://tinyurl.com/o7s3o7t
http://tinyurl.com/nq4nzko )
def.
min poly of AB = x(x-9) ===== m(x)
Let m'(x) denote the minimal polynomial of BA.
Note (AB)^{k+1}=A (BA)^k B and
hence AB p(AB) = A p(BA) B for any polynomial p(x).
Therefore m(AB)=0 implying BA m(BA)=0.
We see that m'(x)| x m(x)=x^2 (x-9).
On the other hand, by Cayley–Hamilton theorem, m'(x) | (x-9)^2
(cf. http://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem )
It turns out that m'(x) is a common factor of x^2 (x-9) and (x-9)^2.
That is, m'(x)=1 or (x-9).
Clearly, m'(x) is not equal to 1,
otherwise BA=I_2, tr(BA)=2, a contradiction with tr(AB)=18.
(cf. http://en.wikipedia.org/wiki/Trace_(linear_algebra) tr(AB)=tr(BA) )
Hence m'(x)=x-9, namely BA=9I_2.
(Remark. Indeed, x-9 must divide m'(x).
cf. http://mathworld.wolfram.com/MatrixMinimalPolynomial.html )
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※ 編輯: Eeon (219.70.173.141), 10/11/2014 04:22:24
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