Why is a projection matrix of an orthogonal projection symmetric?

I am quite new to this, so I hope you forgive me if the question is naïve. (Context: I am learning econometrics from Davidson & MacKinnon’s book “Econometric Theory and Methods”, and they do not seem to explain this; I’ve also looked at Luenberger’s optimization book that deals with projections at an a bit more advanced level, but with no luck).

Suppose that I have an orthogonal projection P with is associated projection matrix P. I am interested in projecting each vector in Rn into some subspace ARn.

Question: why does it follow that P=PT, that is, P is symmetric? What textbook could I look at for this result?

Answer

This is a fundamental results from linear algebra on orthogonal projections. A relatively simple approach is as follows. If u1,,um are orthonormal vectors spanning an m-dimensional subspace A, and U is the n×p matrix with the ui‘s as the columns, then
P=UUT.
This follows directly from the fact that the orthogonal projection of x onto A can be computed in terms of the orthonormal basis of A as
mi=1uiuTix.
It follows directly from the formula above that P2=P and that PT=P.

It is also possible to give a different argument. If P is a projection matrix for an orthogonal projection, then, by definition, for all x,yRn
PxyPy.
Consequently,
0=(Px)T(yPy)=xTPT(IP)y=xT(PTPTP)y
for all x,yRn. This shows that PT=PTP, whence
P=(PT)T=(PTP)T=PTP=PT.

Attribution
Source : Link , Question Author : weez13 , Answer Author : whuber

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