[Math] Why are orthogonal projection matrices not … orthogonal

linear algebramatricesorthogonalityprojective-geometry

I know that given an orthogonal matrix U, then orthogonal projection onto the column space of U is represented by the matrix $UU^t$, which is again orthogonal. I've computed these types of matrices many times now.

But when reading online sources such as Wolfram, they give examples of orthogonal projection matrices with a zero column or a zero row, and a couple of 1s – such as the well-known matrix that projects (x,y,z) to (x,y,0).

But this matrix doesn't even have full rank, let alone be unitary or orthogonal.

Where's my conceptual mistake?


Best Answer

You seem not to have noticed a few things about those calculations you say you've done many times. Given an orthogonal matrix $U$, in fact $UU^T$ is the identity matrix. So yes it is "again orthogonal", but that's a curious way to put it. And yes it is the projection onto the column space of $U$, because in fact that column space is all of $\Bbb R^n$.

On the other hand, if $P$ is the matrix that gives the orthogonal projection onto a proper subspace $V$ of $\Bbb R^n$ then $P$ cannot be orthogonal. An orthogonal matrix maps $\Bbb R^n$ onto itself, while $P$ maps $\Bbb R^n$ onto $V$. So the rank of $P$ should be the dimension of $V$, which is less than $n$.

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