Thank you very much for your answer, but what if I need the eigenvalues in terms of a,b,c,d and not the individual components of a,b,c,d? What I mean is

M={{0,b},{c,d}}

with the eigenvalues

eigenvalues=Eigenvalues[M]

which is equal to:

{1/2 (d - Sqrt[4 b c + d^2]), 1/2 (d + Sqrt[4 b c + d^2])}

Where I now need to expand one of the eigenvalues in a taylor series,

Normal[Simplify[
  Series[eigenvalues[[2]], {b, 0, 1}, {c, 0, 1}], {b > 0, c > 0, 
   d > 0}]]

which has the result:

(b c)/d + d

But here and also when I do this for more complicated matrices, the answer is expressed in the wrong order, because mathematica does not know that b,c,d is matrices and does not commute. What I need is the answer in terms of b,c,d but in the correct order as it would be if b,c,d are matrices.