[Math] Projection matrices

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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?

Best Answer

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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