Properties of symmetric projection matrices 
A square matrix $P$ is called a symmetric projection matrix if $P = P^T$ and $P ^2 = P$. Show that a symmetric projection matrix $P$ satisfies the following properties.

*

*$\|x\|^2=\|Px\|^2+\|(1-P)x\|^2$ for all $x$


*$P$ is positive semidefinite


Note that for a vector $x$,
$$||x||^2 = x^Tx$$
Hence,


*

*is equivalent to


$$x^Tx = (Px)^T(Px) + [(I-P)(x)]^T[(I-P)(x)]$$
Pf:
$$RHS = x^TP^TPx + x^T(I-P)^T(I-P)x$$
$$= x^TPx + x^T(I-P)x$$
$$= x^T[P+(I-P)]x$$
$$= LHS$$
QED



*Suppose $P \in \mathbb R^{n \times n}$


TS: $$x^TPx \ge 0 \ \forall \ x \in \mathbb R^n$$
Pf:
$$x^TPx = x^TPPx = x^TP^TPx=(Px)^T(Px) = ||Px||^2 \ge 0$$
QED
P.S. Apparently all but 1 symmetric projection matrices are not positive definite.
