Suppose we have a symmetric $n \times n$ matrix $A$ with $A^{2}=A$, we want to figure out if the linear transformation $T(\vec{x})=A\vec{x}$ necessarily the orthogonal projection onto a subspace of $\mathbb{R}^{n}$.
For this I obtained two eigenvalues, $\lambda=1$ and $\lambda=0$. Onto which subspace is it an orthogonal projection, and why?
Best Answer
You can write
$$x = (x-Ax) + Ax$$
Now you see that $A(x-Ax)=Ax-A^2x = 0$. While you have $A(Ax)=A^2x= Ax$.
From that, you can conclude that: