Re: [線代] 證明題

[JohnMash (Paul)]

※ 引述《wakke (合理化勒索)》之銘言：
: 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


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◆ From: 112.104.169.48
※ 編輯: JohnMash        來自: 112.104.169.48       (07/14 21:46)
推 profyang    :推~想了一下才懂
推 profyang    :等等 所以你這樣等於也證明了n必須是五的倍數?
→ Vulpix      :確實會有 5|n 這個結論沒錯
推 ppia        :x^4+x^3+x^2+x+1 is irred. in |R[x] ( Eisenstein
      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





→ ppia        :criterion ), so the minimal poly. of A must be
→ ppia        :(i) x-1; (ii) x^4+x^3+x^2+x+1 (iii) x^5-1.
→ ppia        :The char. poly. is : (i) (x-1)^n;
→ ppia        :(ii) (x^4+x^3+x^2+x+1)^(n/4)
→ ppia        :(iii) (x^4+x^3+x^2+x+1)^a (x-1)^b
→ ppia        :a, b are positive integers with 4a+b = n
→ ppia        :Since trA = 0, (iii) with a=b is the only
→ ppia        :possibility. Therefore, 5|n and 1 is an eigenvle

※ 編輯: JohnMash        來自: 112.104.98.253       (07/17 23:08)