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

Mathematics Asked by johnathan on December 23, 2021

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

One 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.

Answered by ad2004 on December 23, 2021

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