# Why Mathematica is treating the product of (specified) real variable as complex?

For the given matrix

mat = MatrixForm[{{2*g*a*e, 0, 2*g*a*d, 0, -2*g*a*e, 0,
g*(-2*c*d - (b*e)/E^(I*f) - E^(I*f)*b*e),
g*((I*b*e)/E^(I*f) - I*E^(I*f)*b*e),
g*((-E^((-I)*f))*b*d - E^(I*f)*b*d + 2*c*e)},
{0, -2*g*a*e, 0, -2*g*a*e, 0, -2*g*a*d,
g*((I*b*e)/E^(I*f) - I*E^(I*f)*b*e),
g*(-2*c*d + (b*e)/E^(I*f) + E^(I*f)*b*e),
g*((I*b*d)/E^(I*f) - I*E^(I*f)*b*d)},
{2*g*a*c, 0, g*((a*b)/E^(I*f) + E^(I*f)*a*b), 0, -2*g*a*c,
g*((I*a*b)/E^(I*f) - I*E^(I*f)*a*b),
g*((-E^((-I)*f))*b*c - E^(I*f)*b*c + 2*d*e),
g*((I*b*c)/E^(I*f) - I*E^(I*f)*b*c), g*(a^2 - b^2 + c^2 + d^2 - e^2)}}]


simplifying the singular values treats the products like a*b, b*c etc as complex even when they are defined as real. How to sort this issue?

 T1[a_, b_, c_, d_, e_, f_, g_] = Simplify[SingularValueList[mat][[1]][[1]],
Assumptions -> Element[a, Reals] && Element[b, Reals] &&
Element[c, Reals] && Element[d, Reals] && Element[e, Reals] &&
Element[a*b, Reals] && Element[a*c, Reals] && Element[a*d, Reals] &&
Element[a*e, Reals] && Element[b*c, Reals] && Element[b*d, Reals] &&
Element[b*e, Reals] && Element[c*d, Reals] && Element[c*e, Reals] &&
Element[d*e, Reals] && 0 < f < Pi && 0 < g < 1]


Which gives for T1[a, b, c, d, 0, \[Pi]/2, g] the following:

 Root[-16*a^8*d^4*g^6 + 16*a^4*b^4*d^4*g^6 - 192*a^6*c^2*d^4*g^6 +
32*a^2*b^4*c^2*d^4*g^6 - 352*a^4*c^4*d^4*g^6 + 16*b^4*c^4*d^4*g^6 -
192*a^2*c^6*d^4*g^6 - 16*c^8*d^4*g^6 - 32*a^6*d^6*g^6 -
96*a^4*c^2*d^6*g^6 - 96*a^2*c^4*d^6*g^6 - 32*c^6*d^6*g^6 -
16*a^4*d^8*g^6 - 32*a^2*c^2*d^8*g^6 - 16*c^4*d^8*g^6 -
128*a^4*b*c^2*d^3*g^6*Conjugate[b*d] - 128*a^2*b*c^4*d^3*g^6*
Conjugate[b*d] - 32*a^5*d^3*g^6*Conjugate[a*b]*Conjugate[b*d] -
32*a^3*b^2*d^3*g^6*Conjugate[a*b]*Conjugate[b*d] -
64*a^3*c^2*d^3*g^6*Conjugate[a*b]*Conjugate[b*d] -
32*a*b^2*c^2*d^3*g^6*Conjugate[a*b]*Conjugate[b*d] -
32*a*c^4*d^3*g^6*Conjugate[a*b]*Conjugate[b*d] -
32*a^3*d^5*g^6*Conjugate[a*b]*Conjugate[b*d] -
32*a*c^2*d^5*g^6*Conjugate[a*b]*Conjugate[b*d] -
32*a^4*c*d^3*g^6*Conjugate[b*c]*Conjugate[b*d] -
32*a^2*b^2*c*d^3*g^6*Conjugate[b*c]*Conjugate[b*d] -
64*a^2*c^3*d^3*g^6*Conjugate[b*c]*Conjugate[b*d] -
32*b^2*c^3*d^3*g^6*Conjugate[b*c]*Conjugate[b*d] -
32*c^5*d^3*g^6*Conjugate[b*c]*Conjugate[b*d] -
32*a^2*c*d^5*g^6*Conjugate[b*c]*Conjugate[b*d] -
32*c^3*d^5*g^6*Conjugate[b*c]*Conjugate[b*d] +
(8*a^6*d^2*g^4 - 8*a^4*b^2*d^2*g^4 + 88*a^4*c^2*d^2*g^4 -
16*a^2*b^2*c^2*d^2*g^4 + 88*a^2*c^4*d^2*g^4 - 8*b^2*c^4*d^2*g^4 +
8*c^6*d^2*g^4 + 32*a^4*d^4*g^4 - 8*a^2*b^2*d^4*g^4 +
64*a^2*c^2*d^4*g^4 - 8*b^2*c^2*d^4*g^4 + 32*c^4*d^4*g^4 +
8*a^2*d^6*g^4 + 8*c^2*d^6*g^4 + 16*a^3*b*d^2*g^4*Conjugate[a*b] +
16*a*b*c^2*d^2*g^4*Conjugate[a*b] + 16*a^2*b*c*d^2*g^4*
Conjugate[b*c] + 16*b*c^3*d^2*g^4*Conjugate[b*c] +
32*a^2*b*c^2*d*g^4*Conjugate[b*d] + 16*a^2*b*d^3*g^4*Conjugate[b*d] +
16*b*c^2*d^3*g^4*Conjugate[b*d] + 8*a^3*d*g^4*Conjugate[a*b]*
Conjugate[b*d] + 8*a*b^2*d*g^4*Conjugate[a*b]*Conjugate[b*d] +
8*a*c^2*d*g^4*Conjugate[a*b]*Conjugate[b*d] +
8*a*d^3*g^4*Conjugate[a*b]*Conjugate[b*d] +
8*a^2*c*d*g^4*Conjugate[b*c]*Conjugate[b*d] +
8*b^2*c*d*g^4*Conjugate[b*c]*Conjugate[b*d] +
8*c^3*d*g^4*Conjugate[b*c]*Conjugate[b*d] +
8*c*d^3*g^4*Conjugate[b*c]*Conjugate[b*d])*#1 +
((-a^4)*g^2 + 2*a^2*b^2*g^2 - b^4*g^2 - 10*a^2*c^2*g^2 +
2*b^2*c^2*g^2 - c^4*g^2 - 10*a^2*d^2*g^2 + 2*b^2*d^2*g^2 -
10*c^2*d^2*g^2 - d^4*g^2 - 4*a*b*g^2*Conjugate[a*b] -
4*b*c*g^2*Conjugate[b*c] - 4*b*d*g^2*Conjugate[b*d])*#1^2 + #1^3 & ,
1]


You are probably not using Mathematica correctly. First, you shouldn't put MatrixForm into SingularValueList. Second, the way you defined your matrix and function T1 does not work for properly applying the parameters. Lastly, use ComplexExpand to simplify the expression, assuming all variables are real, or Simplify/Refine.

Clear[T1, mat];

mat[a_, b_, c_, d_, e_, f_,
g_] := {{2*g*a*e, 0, 2*g*a*d, 0, -2*g*a*e, 0,
g*(-2*c*d - (b*e)/E^(I*f) - E^(I*f)*b*e),
g*((I*b*e)/E^(I*f) - I*E^(I*f)*b*e),
g*((-E^((-I)*f))*b*d - E^(I*f)*b*d + 2*c*e)}, {0, -2*g*a*e,
0, -2*g*a*e, 0, -2*g*a*d, g*((I*b*e)/E^(I*f) - I*E^(I*f)*b*e),
g*(-2*c*d + (b*e)/E^(I*f) + E^(I*f)*b*e),
g*((I*b*d)/E^(I*f) - I*E^(I*f)*b*d)}, {2*g*a*c, 0,
g*((a*b)/E^(I*f) + E^(I*f)*a*b), 0, -2*g*a*c,
g*((I*a*b)/E^(I*f) - I*E^(I*f)*a*b),
g*((-E^((-I)*f))*b*c - E^(I*f)*b*c + 2*d*e),
g*((I*b*c)/E^(I*f) - I*E^(I*f)*b*c),
g*(a^2 - b^2 + c^2 + d^2 - e^2)}};

T1[a_, b_, c_, d_, e_, f_, g_] :=
First@SingularValueList[mat[a, b, c, d, e, f, g]]

T1[a, b, c, d, 0, \[Pi]/2, g]
(* Sqrt[4 a d g Conjugate[a d g] + 4 c d g Conjugate[c d g]] *)

Refine[%, {a, b, c, d, e, f, g} \[Element] Reals]
(* Sqrt[4 a^2 d^2 g^2 + 4 c^2 d^2 g^2] *)

Simplify[%, Assumptions -> {a, b, c, d, e, f, g} \[Element] Reals]
(* 2 Sqrt[a^2 + c^2] Abs[d] Abs[g] *)