Complex conjugation operator

1. Mar 6, 2017

MrRobot

1. The problem statement, all variables and given/known data
Hi, so I have been given the following operator in terms of 3 orthonormal states |Φi>

A = |Φ2><Φ2| + |Φ3><Φ3| - i|Φ1><Φ2| - |Φ1><Φ3| + i|Φ2><Φ1| - |Φ3><Φ1|
So I need to determine whether A is unitary and/or Hermitian and/or a projector and then calculate the eigenvalues and eigenfunctions in the |Φi> basis.

The second question is to find eigenvalues and eigenfunctions of the complex conjugation operator acting on complex functions, Cα(x) = α*(x)

2. Relevant equations

3. The attempt at a solution
So for the first one I said it is an operator because, it cannot be unitary since AAτ ≠ unit matrix and not hermitian since A ≠ A, but now I fail to show A2 = A in order to prove that it is actually a projector. please help if there is an easier way.

The second part of the question am just failing to use that A in the formula A|Φi> = a|Φi> to find the eigenvalues and eigenfunctions.

The second question I don't know where to even start.

2. Mar 6, 2017

Orodruin

Staff Emeritus