Define
$ \mathbf{H}=\mathbf{X}\left(\mathbf{X}^{\prime}\mathbf{X}\right)^{-1}\mathbf{X}^{\prime} $
where $\mathbf{X}$ is of order $n \times k$
and
$ \overline{\mathbf{J}}=\frac{1}{n}\mathbf{J}=\frac{1}{n}\mathbf{1}\mathbf{1}^{\prime} $
where $\mathbf{1}$ is a unit vector of order $n \times 1$.
Now $ \mathbf{H}\mathbf{H}=\mathbf{H} $
and
$ \overline{\mathbf{J}}\overline{\mathbf{J}}=\overline{\mathbf{J}} $
Thus both $\mathbf{H}$ and $\overline{\mathbf{J}}$ are idempotent matrices.
My question is whether $\mathbf{H}-\overline{\mathbf{J}}$ would be idempotent. If so then
$ \left(\mathbf{H}-\mathbf{\overline{\mathbf{J}}}\right)\mathbf{\left(\mathbf{H}-\mathbf{\overline{\mathbf{J}}}\right)}=\mathbf{H}-\mathbf{H\overline{\mathbf{J}}-\overline{\mathbf{J}}H}+\overline{\mathbf{J}}=\mathbf{H}-\mathbf{\overline{\mathbf{J}}-\overline{\mathbf{J}}}+\overline{\mathbf{J}}=\mathbf{H}-\overline{\mathbf{J}} $
But I'm not able to show that
$ \mathbf{H\overline{\mathbf{J}}=\overline{\mathbf{J}}H}=\overline{\mathbf{J}} $
I'd highly appreciate if you guide me to figure this. Thanks for your time and help.