Given a matrix $\mathbf{X}\in\mathbb{C}^{m\times 2}_{2}$, a matrix of full column rank, which of the following elements completes a valid equality $$ \mathbf{I}_{m} - \mathbf{X} \left( \mathbf{X}^{*} \mathbf{X} \right)^{-1} \mathbf{X}^{*} = ? $$
a) $\mathbf{I}_{m}$
b) $\mathbf{0}$
c) $\mathbf{M}$
d) None of the options will work.
I think the answer is $0$, since $ \left( \mathbf{X}^{*} \mathbf{X} \right)^{-1}= \mathbf{X}^{−1}\left( \mathbf{X}^{*} \right)^{−1}$
But the answer is c) M ???
Which of the following is $M$? And the the answer is $M$? Sounds tautological...
Anyway, $M - I - X'(X'X)^{-1}X'$ is the so called annihilator matrix. That is, $M$ is orthogonal projection onto $C(X)^\perp$ i.e., $$ My = y - Hy= y - \hat{y} = e . $$ Your logic is falls as, in general, $n >2$, thus $X$ is non square matrix, hence $X^{-1}$ is undefined in the straightforward way and $\exists (X'X)^{-1}$ iff $rank(X'X)=2$.