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If $W$ is a real matrix and we want the orthogonal projection of some vector $v$ onto $span(W)$, the formula I can find is:

$proj_{span(W)}(v)=W(W^TW)^{-1}W^Tv$

Now, if the columns of $W$ are orthonormal, I can understand the simpler version of the formula:

$proj_{span(W)}(v)=WW^Tv$

Since $W^Tv$ is the dot product of $v$ with each column of $W$ and then left multiplying by $W$ gives the appropriate linear combination of the columns of $W$.

So, my feeling is that in the general formula, $(W^TW)^{-1}$ must represent some kind of way of correcting for the lack of orthonormality in the columns of $W$.

If the columns of $W$ are orthongonal, but not normal, I can see that $W^TW$ would be a diagonal matrix with each column being $||W_{.,i}||^2$ and the inverse of this would be the reciprocal of these which cancels out multiplying by $|W_{.,i}||$ twice with the other matrices in the formula.

But, I can't see how take this one step further. Each entry in $W^TW$ is the dot product of two columns and so I can imagine that inverting this sort of compensates for the columns not being orthonomal, but I can't see how to really capture that mathematically.

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