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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?

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    $\begingroup$ By definition, a linear transformation satisfying $T \circ T = T$ is the projection on the subspace $\lbrace x \text{ } | \text{ }Tx=x \rbrace$. $\endgroup$ – TheSilverDoe Sep 3 '20 at 16:02
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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:

  • The image of $A$ is the subspace on which you project.
  • The orthogonal direction of the subspace on which you project is the image of the linear transformation $Id-A$.
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