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:


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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.