- #1

neomasterc

- 5

- 0

## Homework Statement

Prove I=T

_{1}+T

_{2}+...+T

_{k}

Where T

_{i}=p

_{i}(T)

## Homework Equations

T is kxk

pi(x)=(x-c

_{1})...(x-c

_{k}) is the minimal polynomial of T.

p

_{i}=[itex]\pi[/itex]

_{i}(x)/[itex]\pi[/itex]

_{i}(c

_{i})

[itex]\pi[/itex]

_{i}=[itex]\pi[/itex](x)/(x-c

_{i})

To evaluate these functions at a matrix, simply let c

_{i}=c

_{i}I

## The Attempt at a Solution

From lagrange interpolation, f=Ʃf(x)p

_{i}(x)

so 1=Ʃp

_{i}(x)