How Can Dirac Notation Be Used to Determine Eigenvalues and Eigenfunctions?

[phys]

Homework Statement

I have the following question (see below)

Homework Equations

The eigenvalue equation is Au = pu where u denotes the eigenstate and p denotes the eigenvalue

The Attempt at a Solution

I think that the eigenvalues are +1 and - 1, and the states are (phi + Bphi) and (phi-Bphi)
however I got this by just substituting these in from the symmetry of the operator.

Is there are neat algebraic way to work out the eigenvalues and eigenfunctions as opposed to just substitution?

I am stuck on working out how to express the 0 eigenvalue eigenstates in terms of the projection operator as well ...

Thank you very much

 

[eigenmax]

About the projection operator , I'm not very well versed in them but if |x> is an eigenvector of it's projection operator with an eigenvalue of 1 ,
|x> <x| |x> = |x>
and any vector orthogonal to the |x> shown above is an eigenvector with eigenvalue zero. Also if
|x> <x| |A> = |x> <|A>
then if A is our zero eigenstate it seems a little meaningless.
Not sure how this fits in with a 0 value eigenstate though.
Sorry if I'm wrong.