# linear algebra: show that \$ mathbf {(H- frac {1} {n} J_n)} \$ is indelible

I'm trying to show that matrix. $$mathbf {(H- frac {1} {n} J_n)}$$ It is idempotent where $$mathbf {H}$$ It is the matrix of the linear regression hat and $$J_n$$ is the $$n times n$$ matrix with $$1$$ In all your entries. 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 $$mathbf {H}$$ Y $$mathbf { frac {1} {n} J_n}$$ They are identity content, therefore:

$$mathbf {(H- frac {1} {n} J_n) (H- frac {1} {n} J_n) = HH frac {1} {n} J_n – frac {1} {n } J_nH + frac {1} {n ^ 2} J_n}$$

How would you continue now to show that the given matrix is ​​an identity content?