Properties of symmetric projection matrices

Question

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.

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

2. $P$ is positive semidefinite

Answer 1

Let $\langle -,- \rangle$ be the standard dot product.

Hint: For the first part try writing both sides in terms of the dot product and expanding using linearity. Notice $\langle x, Px \rangle = \langle Px, x \rangle$ since $P$ is symmetric.

For the second part you can use a similar trick to above, this time replacing the $P$ in the dot product you get with $P^2$.

Answer 2

Note that for a vector $x$,

$$||x||^2 = x^Tx$$

Hence,

1. 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

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2. 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.