作者JohnMash (Paul)
看板Math
标题Re: [线代] 证明题
时间Thu Jul 14 19:12:16 2011
※ 引述《wakke (合理化勒索)》之铭言：
: Let A be an n*n matrix over R such that A^5=I .Here, as usual,
: I denotes the identity matrix.
: Show that if the trace tr(A)=0 , then Av=v holds,
: for some non-zero vector v
Factorize x^5-1 in R[x]
x^5-1=(x-1)(x^4+x^3+x^2+x+1)
however, x^4+x^3+x^2+x+1 has NO real roots
hence, x^4+x^3+x^2+x+1=(x^2+ax+b)(x^2+cx+d)
where a=-e^{2πi/5}-e^{-2πi/5}=-2cos(2π/5)
c=-e^{4πi/5}-e^{-4πi/5}=-2cos(4π/5)=2cos(π/5)
and x^5-1=(x-1)(x^2+ax+b)(x^2+cx+d)
Denote the characteristic polynomial of A as f(x)
then f(x) should be (x-1)^k (x^2+ax+b)^p (x^2+cx+d)^q
and k,p,q≧0, k+2p+2q=n and k,p,q are INTEGERS
First, assume p≧q, p=q+r
f(x)=(x-1)^k (x^4+x^3+x^2+x+1)^q (x^2+ax+b)^r
=x^n + (-k+q+ra) x^{n-1} +...
Because, tr(A)=0,
-k+q+ra=0, and a is IRRATIONAL
hence, r=0
k=q and k+4q=n then k≠0
Similarly, if q≧p we also have k≠0
Done
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 112.104.169.48
※ 编辑: JohnMash 来自: 112.104.169.48 (07/14 21:46)
1F：推 profyang :推~想了一下才懂 07/15 00:30
2F：推 profyang :等等 所以你这样等於也证明了n必须是五的倍数? 07/15 00:41
3F：→ Vulpix :确实会有 5|n 这个结论没错 07/15 00:44
4F：推 ppia :x^4+x^3+x^2+x+1 is irred. in |R[x] ( Eisenstein 07/17 19:40
x^4+x^3+x^2+x+1 is irreducible in Q[x]
but reducible in R[x]
x^5-1=(x-1)(x^2+ax+b)(x^2+cx+d)
In mathematics, Eisenstein's criterion gives an easily checked sufficient
condition for a polynomial with integer coefficients to be irreducible over
the rational numbers.
http://en.wikipedia.org/wiki/Eisenstein's_criterion
5F：→ ppia :criterion ), so the minimal poly. of A must be 07/17 19:40
6F：→ ppia :(i) x-1; (ii) x^4+x^3+x^2+x+1 (iii) x^5-1. 07/17 19:41
7F：→ ppia :The char. poly. is : (i) (x-1)^n; 07/17 19:42
8F：→ ppia :(ii) (x^4+x^3+x^2+x+1)^(n/4) 07/17 19:43
9F：→ ppia :(iii) (x^4+x^3+x^2+x+1)^a (x-1)^b 07/17 19:44
10F：→ ppia :a, b are positive integers with 4a+b = n 07/17 19:44
11F：→ ppia :Since trA = 0, (iii) with a=b is the only 07/17 19:44
12F：→ ppia :possibility. Therefore, 5|n and 1 is an eigenvle 07/17 19:45
※ 编辑: JohnMash 来自: 112.104.98.253 (07/17 23:08)