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I am dealing with the situation where despite of specifying my variables b1 and c to be greater than zero, Mathematica still returns the output with Re[b1+c]

x = a - (b1 + c)/2 I

FullSimplify[
 Im[x],
 {a >= 0, b1 >= 0, c >= 0}
 ]
(* -(1/2) Re[b1 + c] *)
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  • $\begingroup$ Next time please take the time to contsruct a minimal example and use readable formatting in your post. See my edit for reference. $\endgroup$
    – Szabolcs
    Jun 18, 2021 at 9:13

2 Answers 2

Reset to default
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Clear["Global`*"]

mat1 = Simplify[({{a - (b1 + c)/2 I, d}, {d, a - (b2 - c)/2 I}})];

mat2 = ({{0, 0, 1, 0}, {0, 0, 0, 1}, {-1, 0, 0, 0}, {0, -1, 0, 0}});

The primary tool for simplifying complex expressions is ComplexExpand

MatrixForm[
 Simplify@
  ComplexExpand[({{Re[mat1[[1]][[1]]], 
      Re[mat1[[1]][[2]]], -Im[mat1[[1]][[1]]], -Im[mat1[[1]][[2]]]}, {Re[
       mat1[[2]][[1]]], 
      Re[mat1[[2]][[2]]], -Im[mat1[[2]][[1]]], -Im[mat1[[2]][[2]]]}, {Im[
       mat1[[1]][[1]]], Im[mat1[[1]][[2]]], Re[mat1[[1]][[1]]], 
      Re[mat1[[1]][[2]]]}, {Im[mat1[[2]][[1]]], Im[mat1[[2]][[2]]], 
      Re[mat1[[2]][[1]]], Re[mat1[[2]][[2]]]}})]]

enter image description here

This result shows that you need only require that the variables are real rather than the more restrictive nonnegative.

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  • $\begingroup$ I think it's worth noting that while Simplify and Refine are "safe", in that they shouldn't make invalid assumptions, ComplexExpand assumes that variables are real, unless told otherwise. I treat ComplexExpand (and PowerExpand) as unsafe, that is every use needs careful examination to ensure that their assumptions are met. $\endgroup$
    – mikado
    Jun 17, 2021 at 22:20
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I don't know really know why FullSimplify doesn't work, but Refine does

Assuming[{a >= 0, d >= 0, b1 >= 0, b2 >= 0, c >= 0}, 
 Refine[({{Re[mat1[[1]][[1]]], 
     Re[mat1[[1]][[2]]], -Im[mat1[[1]][[1]]], -Im[
       mat1[[1]][[2]]]}, {Re[mat1[[2]][[1]]], 
     Re[mat1[[2]][[2]]], -Im[mat1[[2]][[1]]], -Im[
       mat1[[2]][[2]]]}, {Im[mat1[[1]][[1]]], Im[mat1[[1]][[2]]], 
     Re[mat1[[1]][[1]]], Re[mat1[[1]][[2]]]}, {Im[mat1[[2]][[1]]], 
     Im[mat1[[2]][[2]]], Re[mat1[[2]][[1]]], Re[mat1[[2]][[2]]]}})]]
(* {{a, d, (b1 + c)/2, 0}, {d, a, 0, (b2 - c)/2}, {1/2 (-b1 - c),
   0, a, d}, {0, 1/2 (-b2 + c), d, a}} *)

Occasionally, a more specific simplification, such as Refine finds what you want when the more general one, FullSimplify, does not. Presumably FullSimplify tries a lot of other things before it gives up.

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