# Why is Mathematica not simplifying this matrix?

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?

• Try ComplexExpand. Commented May 12 at 18:02

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.

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])}}*)