Why is Mathematica not simplifying this matrix?

Question

I have trouble simplifying the following matrix

A = 
{{(a^2 + b^2 + c^2 + Im[Sqrt[(a - I*b)^2 + c^2]]^2 + 
    Re[Sqrt[(a - I*b)^2 + c^2]]^2)/
   (2*Sqrt[(a - I*b)^2 + c^2]*Conjugate[
     Sqrt[(a - I*b)^2 + c^2]]), 
  (I*b*c)/(Sqrt[(a - I*b)^2 + c^2]*Conjugate[
     Sqrt[(a - I*b)^2 + c^2]])}, 
 {(-((a + I*b)*(a^2 + b^2 + c^2)) + 
    (a - I*b)*Conjugate[Sqrt[(a - I*b)^2 + c^2]]^2)/
   (2*c*Sqrt[(a - I*b)^2 + c^2]*Conjugate[
     Sqrt[(a - I*b)^2 + c^2]]), 
  (Sqrt[(a - I*b)^2 + c^2] + (2*b*((-I)*a + b))/
     Conjugate[Sqrt[(a - I*b)^2 + c^2]] + 
    Conjugate[Sqrt[(a - I*b)^2 + c^2]])/
   (2*Sqrt[(a - I*b)^2 + c^2])}}; 

Here, $a,b,c$ are all real and positive. Firstly, Im[Sqrt[(a - I*b)^2 + c^2]]^2 + Re[Sqrt[(a - I*b)^2 + c^2]]^2 should be just Abs[Sqrt[(a - I*b)^2 + c^2]]^2. Secondly, when checked numerically, the diagonal elements seems to be identical. How can I simplify this matrix in the best possible way?

Answer 1

I don't know if this helps, but

f[e_] := 100 Count[e, _Conjugate, {0, Infinity}] + 100 Count[e, _Sign, {0, Infinity}] + LeafCount[e];
FullSimplify[A, Assumptions -> {a > 0, b > 0, c > 0}, ComplexityFunction -> f]

yields

{{1/2 (1 + (a^2 + b^2 + c^2)/Abs[(a - I b)^2 + c^2]), (I b c)/
  Abs[(a - I b)^2 + c^2]}, {-((I b c)/Abs[(a - I b)^2 + c^2]), (
  2 b (-I a + b) + Abs[(a - I b)^2 + c^2] + 
   Conjugate[Sqrt[(a - I b)^2 + c^2]]^2)/(2 Abs[(a - I b)^2 + c^2])}}

which shows that the off-diagonal elements appear to be opposite sign.

Answer 2

To simplify objects with complex numbers, we can use ComplexExpand first.

Assuming[a > 0 && b > 0 && c > 0, A // ComplexExpand // FullSimplify]

(* {{(a^2 + b^2 + c^2 + Sqrt[4 a^2 b^2 + (a^2 - b^2 + c^2)^2])/(
  2 Sqrt[4 a^2 b^2 + (a^2 - b^2 + c^2)^2]), (I b c)/Sqrt[
  4 a^2 b^2 + (a^2 - b^2 + c^2)^2]}, {-((I b c)/Sqrt[
   4 a^2 b^2 + (a^2 - b^2 + c^2)^2]), (
  a^2 + b^2 + c^2 + Sqrt[4 a^2 b^2 + (a^2 - b^2 + c^2)^2])/(
  2 Sqrt[4 a^2 b^2 + (a^2 - b^2 + c^2)^2])}}*)