Htykuut

Please give me some hints for the right direction, but don't give me a full answer.

We define $mathbf{P_X}$ as projection matrices: $mathbf{P_X = X(X'X)^{-1}X'}$.
My exercise reads:
Prove if both $mathbf{A}$ and $mathbf{AB}$ are full column rank, then: $mathbf{P_A - P_{AB}}geq 0$.

First of all, I'm pretty sure the zero should be boldfaced, as I expect to end up with a matrix, not a scalar.
Furthermore, I know $mathbf{P_X} in mathcal{P}$ implies $mathbf{I-P_X} in mathcal{P}$. Hence, we have $mathbf{0 leq P_A leq I}$ and $mathbf{0 leq P_{AB} leq I}$. However, this proof requires me to show $mathbf{P_A geq P_{AB}}$, and I'm not sure how to approach it.

Does it have somehting to do with the ranks of matrices $mathbf{A}$ and $mathbf{AB}$? If so, how does it relate to the "size" of the projection matrices?

The hint that eventually solved it for me was that if you indeed had to show $mathbf{P_A - P_{AB}} geq mathbf{0}$, you could also show $mathbf{P_A - P_{AB}}$ was a projection matrix itself. (Thus: $mathbf{P_A - P_{AB}} in mathcal{P}$)