Projection matrices

Question

I have found these two apparently contradicting remarks about projection matrices:

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

2. $P$ 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?

Answer 1

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.

Answer 2

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.

Answer 3

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.