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Lagrange Interpolation and Matrices

  • Thread starter neomasterc
  • Start date
  • #1

Homework Statement


Prove I=T1+T2+...+Tk
Where Ti=pi(T)

Homework 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

The Attempt at a Solution


From lagrange interpolation, f=Ʃf(x)pi(x)
so 1=Ʃpi(x)
 

Answers and Replies

  • #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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