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?
3 Answers
Let $A:=\pmatrix{1&1\\0&0}$. We have $$A\cdot A=\pmatrix{1&1\\0&0}\cdot\pmatrix{1&1\\0&0}=\pmatrix{1&1\\0&0}=A,$$ but $A$ is not hermitian.
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.
One family of examples of matrices that are idempotent and unsymmetric is given by the $n\times n$ matrices $\frac12(\mathbf I+\mathbf H\mathbf D)$ and $\frac12(\mathbf I-\mathbf H\mathbf D)$, where $\mathbf H$ is an $n\times n$ Hilbert matrix, and $\mathbf D=\mathrm{diag}\left(\left.(-1)^j j \binom{n+j-1}{j-1} \binom{n}{j}\right|_{j=1,\dots,n}\right)$
See Householder and Carpenter for more details.
