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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?

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It'd be interesting, perhaps, to know what they exactly mean by "projectors are always positive"...and why they have two different names to define the same thing: $\,P^2=P\,$ is idempotent (def. 1) and it is a projector (def. 2)...? Anyway, as the example below here shows, a projector is not always Hermitian. – DonAntonio Jul 18 '12 at 13:39
@Euclean, It might be nice to point out to the author of 2) that there is a mistake. – Jonas Meyer Jul 18 '12 at 15:27
Sure! I'll do so, good idea! – Euclean Jul 18 '12 at 15:39
up vote 6 down vote accepted

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.

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Thanks for both examples. I realize now that both statements are not at all similar: in the text corresponding to 2), $P$ is a homomorphism, where in the text corresponding to 1) there is not such a requirement. – Euclean Jul 18 '12 at 14:04
@Euclean: No, 2) is just false as stated, and in both 1) and 2) it is assumed that $P$ is linear. Positivity does not follow from $P^2=P$ as these examples show. – Jonas Meyer Jul 18 '12 at 15:03
@JonasMeyer You are reading my mind, ok I think I'm going to think a little about both questions before replying to you again ;O) – Euclean Jul 18 '12 at 15:05

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.

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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.

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