# Orthogonal Projections- Properties

I was looking for some help on my math homework.

The question is, let $P=P_E$ be the matrix of an orthogonal projection onto a subspace E. Show that:

a. The matrix is self-adjoint, meaning A=A*

b. $P^2=P$

I just cannot really figure out where to start on both parts of these. Does anyone have any suggestions? I see how the properties are applied when looking at orthogonal projection matrices, I just cannot see where to go to start trying to prove it.

• What is your definition of an orthogonal projection? We can work from there. Oct 27, 2016 at 6:52
• The one that I was instructed to you from this problem is that the matrix of orthogonal projection is the summation from k=1 to r of 1/(norm of v$k$ ) squared times v$k$ adjoint times v$k$ Oct 27, 2016 at 6:55
• That last comment doesn't make much sense. You're stating that the matrix of $P$ is a number, it's not, it's a matrix. So again, what is the $ij$-th entry of the matrix $P$? Judging by your comment above, it also depends on a certain basis $\left\{v_1, \dots , v_n\right\}$, what kind of basis? Be precise in your formulations. Oct 27, 2016 at 7:01
• That is a great question. That is our formula we are given in the text book verbatim, sorry that is not helpful. I am also confused by that. I do know we need an orthogonal basis. Oct 27, 2016 at 23:40