Solution
= Solution
Using $M^2=M$,
$$
\boxed{(I-M)^2=I-2M+M^2=I-M.}
$$
Thus $I-M$ is another <idempotent matrix>. In fact $M(I-M)=(I-M)M=0$: as <linear projections>, they select the complementary summands in the decomposition $x=Mx+(I-M)x$.
= Solution
Using $M^2=M$,
$$
\boxed{(I-M)^2=I-2M+M^2=I-M.}
$$
Thus $I-M$ is another <idempotent matrix>. In fact $M(I-M)=(I-M)M=0$: as <linear projections>, they select the complementary summands in the decomposition $x=Mx+(I-M)x$.