If Mu is a projection matrix, how can I show that Mu^2=Mu by direct computation?

Question

The definition of Mu, or the projection matrix, I am working with is Mu(a)=Pu(a). Let's say that a set of vectors (v1, v2.....vm) is an orthonormal basis of U. Then, I can calculate Pu(a) by doing (v1a)v1+(v2a)v2+....(vm*a)vm. Eventually, I would be able to conclude that Mu=VV^T. Am I on the right track? How can I conclude that Mu^2=Mu otherwise? Thanks

ad2004 · Answer

Your approach appears to be on the right track.  The general expression for a projection matrix is:
$$
P=Aleft(A^{T}Aright)^{-1}A^{T}
$$
so from this you can write:
$$
P^{2}=Aleft(A^{T}Aright)^{-1}A^{T}Aleft(A^{T}Aright)^{-1}A^{T}
$$.
Since we get a cancellation due to $left(A^{T}Aright)^{-1}A^{T}A=I$ in the middle, this yields:
$$
P^{2}=Aleft(A^{T}Aright)^{-1}A^{T}=P
$$.
I hope this helps.

If Mu is a projection matrix, how can I show that Mu^2=Mu by direct computation?

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