1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
    pi=[itex]\pi[/itex]i(x)/[itex]\pi[/itex]i(ci)
    [itex]\pi[/itex]i=[itex]\pi[/itex](x)/(x-ci)

    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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Lagrange Interpolation and Matrices
  1. Lagrange Interpolation (Replies: 0)

Loading...