Is this what you want?

With[{b = {{b1}, {b2}}, a = {{a1}, {a2}}, S = {{s11, s12}, {s12, s22}}}, 
 ConjugateTranspose[S.a].(S.a) == ConjugateTranspose[a].a // FullSimplify // PowerExpand]
ComplexExpand[%]

gives

 {{a1^2 s11^2 + 2 a1 a2 s11 s12 + a1^2 s12^2 + a2^2 s12^2 + 2 a1 a2 s12 s22 + a2^2 s22^2}} == {{a1^2 + a2^2}}

Hi,

Sorry for the ambiguity.

I followed your suggested method and rewrote the code as follows:

With[{b = {{b1}, {b2}}, a = {{a1}, {a2}}, 
  S1 = {{s11, s12}, {s12, s22}}},
 rule1 = Conjugate[x_ + y_] :> Conjugate[x] + Conjugate[y]; 
 rule2 = Conjugate[x_ y_] :> Conjugate[x] Conjugate[y]; 
 myRules = {rule1, rule2}; 
 ConjugateTranspose[S1.a]. (S1.a) == ConjugateTranspose[a].a //. 
    myRules // Expand // TraditionalForm]

So I got the following result:

({
   {a1 s11 Conjugate[a1] Conjugate[s11] + 
     a2 s12 Conjugate[a1] Conjugate[s11] + 
     a1 s12 Conjugate[a1] Conjugate[s12] + 
     a2 s22 Conjugate[a1] Conjugate[s12] + 
     a1 s11 Conjugate[a2] Conjugate[s12] + 
     a2 s12 Conjugate[a2] Conjugate[s12] + 
     a1 s12 Conjugate[a2] Conjugate[s22] + 
     a2 s22 Conjugate[a2] Conjugate[s22]}
  }) == ({
   {a1 Conjugate[a1] + a2 Conjugate[a2]}
  })

So my question is what is the sequence of commands that results in substituting EACH and EVERY multiplication of a complex variable with its conjugate with its square absolute value (for example: a2 s12 Conjugate[a2] Conjugate[s12] should be substituted with Abs[a2]^2 Abs[s12]^2). This can usually be done by using the command FullSimplify. However, this command seems to be selective in this case and it is not substituting all the terms.

Thanks

As far as I know, ComplexExpand assumes that all variables in its argument are real and then expands it accordingly. However, in the case I am asking about, variables may be complex. So what I want is, that Mathematica substitutes (a1 s12 + a2 s22)* with a1* s12* + a2* s22*. This equation is necessary for the S-matrix. With reference to your website, I think that you're quite familiar with it,

Thanks

Sorry but I am not following you and do not know what output you are after now.

How about we rewind and start from the initial state again? Could you show the input and show the output you are looking for. Then one can help write the rules/commands needed to transform the input to the output.