I know that a matrix $P$ is a projection matrix IFF $P=P^2$, but is it also true that it must be such that $P=P^T$?
I thought that was only true for orthogonal projection matrices.
But i am reading https://see.stanford.edu/materials/lsoeldsee263/homeworkProblems.pdf on page 24, exercise 4.3 it says that BOTH must be true for it to be a projection matrix. Is that correct?
Best Answer
It is a definition, so of course one could choose to define a projection with both properties or with just one. I've seen authors use both. Requiring both properties to hold tends to be a more common approach. Consider the matrix $$P=\begin{bmatrix}1&0\\1&0\end{bmatrix}$$ Now $P^2=P$, but $P\ne P^T$. If we consider the action of $P$ on some vector, say $P(2,3)=(2,2)$, then we see that the "error" vector $(0,1)$ is not orthogonal to the projected vector, $(2,2)$. This is a property that is generally desirable for geometric projections, so a common convention requires $P=P^T$ in order for a matrix to qualify as a projection.
But really, it is a matter of convention. Do you, as an author, want your projections to be orthogonal, or not?