I have a question:
Let A be an idempotent matrix.
- Show that I-A is idempotent
- Show that I+A is nonsingular and (I+A)^(-1) = I- ${1\over 2}$A
I solved 1: (I - A)^2
= (I - A)(I - A)
= I^2 - AI - AI + A^2
= I - 2A + A^2
= I - 2A + A since A is idempotent
= I - A.
So it is idempotent
However I don't know how to get startet with 2,
any help is appreciated.
Thanks in advance
$\endgroup$1 Answer
$\begingroup$Note that $$(I + A)(I - (1/2)A)$$ $$= (I + A)I + (I + A)(-1/2)A $$ $$= (I + A) + (-(1/2) A - (1/2) A^2) $$ $$= (I + A) + (-(1/2) A - (1/2) A)$$ (Since A is idempotent) $$= (I + A) - A $$ $$= I$$
Hence, $(I + A)$ is nonsingular, with $(I + A)^{-1} = I - (1/2)(A)$ by the uniqueness of inverses.
$\endgroup$ 6
