Lagrange Interpolation and Matrices

[neomasterc]

Homework Statement

Prove I=T₁+T₂+...+T_k
Where T_i=p_i(T)

Homework Equations

T is kxk
pi(x)=(x-c₁)...(x-c_k) is the minimal polynomial of T.
p_i=\pi_i(x)/\pi_i(c_i)
\pi_i=\pi(x)/(x-c_i)

To evaluate these functions at a matrix, simply let c_i=c_iI

The Attempt at a Solution

From lagrange interpolation, f=Ʃf(x)p_i(x)
so 1=Ʃp_i(x)

 

[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