# Show if a linear transformation necessarily the orthogonal projection

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?

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

$$x = (x-Ax) + Ax$$
Now you see that $$A(x-Ax)=Ax-A^2x = 0$$. While you have $$A(Ax)=A^2x= Ax$$.
• 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 $$Id-A$$.