Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 30 / 1 / b

Full column matrix rank makes $X^{T}X$ a positive-definite matrix, and its matrix inverse is symmetric. Thus the hat matrix satisfiesAlso $PX=X$ and the image of $P$ is contained in the column space of $X$. These identities show that $P$ is the orthogonal projection matrix onto the column space of $X$.