I am trying to show that the matrix $\mathbf{(H-\frac{1}{n}J_n)}$ is idempotent where $\mathbf{H}$ is the Hat-matrix (Projection matrix) of linear regression and $J_n$ is the $n\times n$ matrix with $1$ in all its inputs. Taking :
$$\mathbf{(H-\frac{1}{n}J_n)(H-\frac{1}{n}J_n)= HH - H\frac{1}{n}J_n - \frac{1}{n}J_nH + \frac{1}{n}J_n\frac{1}{n}J_n}$$
Now, we know that $\mathbf{H}$ and $\mathbf{\frac{1}{n}J_n}$ are idempontent, thus :
$$\mathbf{(H-\frac{1}{n}J_n)(H-\frac{1}{n}J_n)=H-H\frac{1}{n}J_n - \frac{1}{n}J_nH +\frac{1}{n}J_n}$$
How would I continue now in order to show that the given matrix is idempontent ?