TITLE: Projection matrices
QUESTION [6 upvotes]: I have found these two apparently contradicting remarks about projection matrices:

A matrix $P$ is idempotent if $PP = P$. An idempotent matrix that is also Hermitian is called a projection matrix.

$P$ is a projector if $PP = P$. Projectors are always positive which implies that they are always Hermitian.


Which of both is correct? Is a matrix $P$ that verifies $PP=P$ always Hermitian?

REPLY [6 votes]: The fact that a projection matrix is Hermitian or not depends on your definition of projection matrices. Usually, if $P$ satisfies $PP = P$, then $P$ is idempotent, and is called a projection matrix, no matter it's Hermitian or not. If $P$ is also Hermitian, then it's called orthogonal projection, otherwise it's oblique projection. But some authors only define Hermitian idempotent matrix (orthogonal projection) as projection. See here to find more.