# Lagrange Interpolation and Matrices

1. Nov 21, 2011

### neomasterc

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=$\pi$i(x)/$\pi$i(ci)
$\pi$i=$\pi$(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. Nov 21, 2011

### neomasterc

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