Show that the characteristic roots of an idempotent matrix are either zero or unity.

Explanation

Let A be an idempotent matrix, then
`A^(2)=A`
If `lambda` be an eigen value of the matrix A correspondin to eigen vector X,so that
`AX=lambdaX`
where `X!=0`
from Eq. (ii) , `A(AX)=A(lambdaX)`
`rArr (A A)X=lambda(AX)`
` rArr A^(2)x=lambda (lambdaX)`
`rArr AX = lambda^(2)X`
`rArr lambdaX=lambda^(2)X`
`rArr (lambda-lambda^(2))X=0`
`rArr lambda-lambda^(2)=0`
`therefore lambda=0`
or lambda=1`