$P$ is projection matrix iff $A$ is reflection matrix?

Question

I have the following definition

An $n \times n$ matrix $A$ is a reflection matrix if and only if $A^2 = I$ and $A^T= A$. A projection matrix is $P = 1/2(A+I)$.

I was wondering if I can conclude that $P$ is a projection matrix if and only if $A$ is a reflection matrix. If it can be said can you please explain why?

Assuming that this is true can I say that $A^2 = I$ and $A^T = A$ if and only if $P^2 = P$ and $P^T=P$?

Answer 1

No, a projection matrix $P$ satisfies $P=P^2$, and is not symmetric in general. Only orthogonal projection matrices are symmetric. Furthermore $P^2=I$ would imply $P=I$. So symmetric involutory matrices $A$ with $A^2=I$ are something different than projection matrices.

Edit: You have added the relation $P=(A+I)/2$. Then of course we have $$ P^2=(A+I)^2/4=(A^2+2A+I)/4=(I+A)/2=P. $$

Answer 2

Since you’re restricting $P$ to an orthogonal projection, consider one of the standard ways to construct the (orthogonal) reflection of a vector relative to some subspace of $\mathbb R^n$: find the orthogonal rejection of the vector from that subspace and reverse it. That is, if $W\subset\mathbb R^n$ is a subspace and $\pi_W$ is orthogonal projection onto $W$, then the reflection of a vector $v$ in $W$ is $\rho_Wv=\pi_Wv-(v-\pi_wv)=2\pi_Wv-v$, or, in matrix form, $Av=(2P-I)v$, from which your equation $P=(A+I)/2$ follows.