How Do You Determine if an Operator is Unitary, Hermitian, or a Projector?

[MrRobot]

Homework Statement

Hi, so I have been given the following operator in terms of 3 orthonormal states |Φ_i>

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 |Φ_i> basis.

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

Homework Equations

The Attempt at a Solution

So for the first one I said it is an operator because, it cannot be unitary since AA^τ ≠ 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|Φ_i> = a|Φ_i> to find the eigenvalues and eigenfunctions.

The second question I don't know where to even start.
Please help, thank you very much.

 

[Orodruin]

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.

Please provide your actual attempt.

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

What is the definition of an eigenfunction? How does complex conjugation act on a general complex function?