Minimal and characteristic polynomial

  • #1
Let V =Mn(k),n>1 and T:V→V defined by T(M)=Mt (transpose of M).
i) Find the minimal polynomial of T. Is T diagonalisable when k = R,C,F2?
ii) Suppose k = R. Find the characteristic polynomial chT .
I know that T2=T(Mt))=M and that has got to help me find the minimal polynomial
 

Answers and Replies

  • #2
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3,297
You'll need to find a polynomial [tex]p(x)=x^n+...+a_1x+a_0[/tex], such that p(T)=0, i.e.

[tex]T^n+...+a_1T+a_0I=0[/tex]

You know that [tex]T^2(M)=M[/tex], can you use this to find a suitable polynomial??
 
  • #3
All i can think of is f(x)=1 or f(T)=I
 
  • #4
22,129
3,297
Come on, how do you write [tex]T^2(M)=M[/tex] as a polynomial in T??
 

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