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rmatrixprojection

Find projection matrix to create zero-sum vector

I have subspace of vectors w whose elements sum to 0.

I would like to find a projection matrix Z such that it projects any x vector onto the subspace w (i.e., a subspace where the vector sums to 0).

Is there an R function to do this?

Solution

  • The question did not specify how w is provided but if w is a matrix with full rank spanning the space w then 

    Z <- w %*% solve(crossprod(w), t(w))

    If w has orthogonal columns then the above line reduces to:

    Z <- tcrossprod(w)

    Another possibility is to use the pracma package in which case w need not be of full rank:

    library(pracma)
Z <- tcrossprod(orth(w))

    If w were the space of all n-vectors that sum to zero then:

    Z <- diag(n) - matrix(1, n, n) / n

    Note Have revised after re-reading question.