Consider $\mathbf{X} \in \mathbb{R}^{n \times p}$ as the design matrix, which is full rank, and $\mathbf{W} = (\mathbf{I} - \mathbf{1}_n\mathbf{1}_n^T)\mathbf{X}$ as the centered design matrix.
I was wondering if $\mathbf{P_X} = \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T$ is a projection matrix on the column space of $\mathbf{X}$, then would $\mathbf{P_W} = \mathbf{X}(\mathbf{W}^T\mathbf{W})^{-1}\mathbf{X}^T$ be also a projection matrix on to the same column space of $\mathbf{X? }$