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 A⊂Rn.

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

m∑i=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,y∈Rn

Px⊥y−Py.

Consequently,

0=(Px)T(y−Py)=xTPT(I−P)y=xT(PT−PTP)y

for all x,y∈Rn. This shows that PT=PTP, whence

P=(PT)T=(PTP)T=PTP=PT.

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