Eigenvalues/eignevectors of Jones matrix

[Come]

I did an exercice for an optic course and the question was to find which optical component, using eigenvalues and eigenvectors, the following Jones matrix was (the common phase is not considered) :

1 i
i 1

I found that this is a quarter-wave plate oriented at 45 degree from the incident plan. To get this result, I assumed that the eigenvectors (without the normalisation):

1
1

and

1
-1 are giving the orientation (±45 degree) of the optical component, and the eigenvalues :

1+i = √(2)e^i*π/4
and
1-i = √(2)e^-i*π/4

are giving the phase shift.

My question is, how can I justify that this assumption about the eigenvectors giving the 'geometry' and eigenvalues giving the phase shift is correct (I know that this is a quarter-wave plate by testing empiricaly with a linearly polarised incident light).

 

[jbsteele]

I am not familiar with the math behind the Jones calculus and I would like to understand this point better.Thank you