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Lagrange Interpolation and Matrices

  1. Nov 21, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove I=T1+T2+...+Tk
    Where Ti=pi(T)

    2. Relevant equations
    T is kxk
    pi(x)=(x-c1)...(x-ck) is the minimal polynomial of T.

    To evaluate these functions at a matrix, simply let ci=ciI
    3. The attempt at a solution
    From lagrange interpolation, f=Ʃf(x)pi(x)
    so 1=Ʃpi(x)
  2. jcsd
  3. Nov 21, 2011 #2
    nvm I solved it. We see Ti*Tj=0 for i!=j, and Ti^2=Ti. Then we get that T1+T2+...+Tk=I or Ti. But if it equals Ti, then T1+T2+...+Ti-1+Ti+1+...Tk=0, which is false, so T1+T2+...+Tk=I
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