Why is orthogonal projection matrix has property that $P=P^T$

Question

I know that definition of orthogonal projection matrix is its range and null space are orthogonal.

And, definition of projection matrix is $P=P^2$

Then, I understand that if $P$ is $n\times n$ projection matrix. $P=P^T \Rightarrow P$ is orthogonal projection matrix(using fundamental theorem of linear algebra)

Then, how could I proof that $P$ is orthogonal projection matrix $\Rightarrow P=P^T$

Can I solve it with using SVD?

Use orthogonality of Px and y-Py as well as orthogonality of Py and x-Px. — Kavi Rama Murthy, Oct 23 '17 at 06:36

score 3 · Accepted Answer · edited Mar 24 '24 at 18:31

3

For any projection $P$, the kernel of $P$ is the image of $I-P$, and $\ker P \oplus \Im P = V$. So you only need to show that if these two spaces are orthogonal, then $P$ is self adjoint. We can directly check: $$\langle Px, y\rangle = \langle Px, (I-P)y + Py\rangle = \langle Px, Py \rangle, $$ and similarly $\langle x, Py \rangle = \langle Px, Py \rangle $, so we have equality.

edited Mar 24 '24 at 18:31

pbb

58
5

answered Oct 23 '17 at 08:34

orangeskid

53,909

How do you conclude the self-adjointness from this equality? (I know this might be obvious if one knows the basics, but I don't.) – HappyFace Jan 08 '23 at 18:35
https://en.wikipedia.org/wiki/Hermitian_adjoint – HappyFace Jan 22 '23 at 08:17

Why is orthogonal projection matrix has property that $P=P^T$

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