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I intend to simplify the first eigenvalue of a particular matrix since it's an extremely ugly expression. I begin with the following 3x3 matrix

M3 = {{-I*ω + Γ/2, -I*g1, 0}, {-I*g1, -I*ω + κ1/2, -I*g2}, {0, - I*g2, -I*ω + κ2/2}};

Finding eigenvalues

valsM3 = Eigenvalues[M3, Cubics -> True];

Printing (might take a while since I'm using FullSimplify)

valsM3[[1]]//FullSimplify

Gives a long expression

1/12 (-((2 2^(
    1/3) (12 g1^2 + 
      12 g2^2 - Γ^2 + Γ κ1 - \
κ1^2 + (Γ + κ1) κ2 - \
κ2^2))/(-36 g1^2 (Γ + κ1 - 
       2 κ2) + (2 Γ - κ1 - κ2) \
(36 g2^2 + (Γ + κ1 - 
          2 κ2) (Γ - 
          2 κ1 + κ2)) + √(4 (12 g1^2 + 
          12 g2^2 - Γ^2 + Γ κ1 \
- κ1^2 + (Γ + κ1) κ2 - \
κ2^2)^3 + (-36 g1^2 (Γ + κ1 - 
            2 κ2) + (2 Γ - κ1 - \
κ2) (36 g2^2 + (Γ + κ1 - 
               2 κ2) (Γ - 
               2 κ1 + κ2)))^2))^(1/3)) + 
2^(2/3) (-36 g1^2 (Γ + κ1 - 
     2 κ2) + (2 Γ - κ1 - κ2) \
(36 g2^2 + (Γ + κ1 - 
        2 κ2) (Γ - 
        2 κ1 + κ2)) + √(4 (12 g1^2 + 
        12 g2^2 - Γ^2 + Γ κ1 - \
κ1^2 + (Γ + κ1) κ2 - \
κ2^2)^3 + (-36 g1^2 (Γ + κ1 - 
          2 κ2) + (2 Γ - κ1 - \
κ2) (36 g2^2 + (Γ + κ1 - 
             2 κ2) (Γ - 
             2 κ1 + κ2)))^2))^(1/3) + 
2 (Γ + κ1 + κ2 - 6 I ω))

In all its ugliness, there are a few recurring expressions that shows up in the eigenvalue. From here on, I perform repeated simplifications using FullSimplify (I could nest them eventually but I'm doing this for the sake of brevity). Define and compililng

j1 = FullSimplify[valsM3[[1]], s1 == 12 g1^2 + 
12 g2^2 - Γ^2 + Γ κ1 - \
κ1^2 + (Γ + κ1) κ2 - κ2^2]

Returns the simplified eigenvalue in terms of the variable s1

1/12 (-((2 2^(1/3)
     s1)/(-3 s1 (Γ + κ1 - 
       2 κ2) - (Γ + κ1 - 
      2 κ2)^3 + 108 g2^2 (Γ - κ2) + 
    Sqrt[4 s1^3 + 
     9 s1^2 (Γ + κ1 - 2 κ2)^2 + 
     6 s1 (Γ + κ1 - 
        2 κ2) ((Γ + κ1 - 
          2 κ2)^3 + 
        108 g2^2 (-Γ + κ2)) + ((\
Γ + κ1 - 2 κ2)^3 + 
       108 g2^2 (-Γ + κ2))^2])^(1/3)) + 
2^(2/3) (-3 s1 (Γ + κ1 - 
     2 κ2) - (Γ + κ1 - 
    2 κ2)^3 + 108 g2^2 (Γ - κ2) + 
  Sqrt[4 s1^3 + 
   9 s1^2 (Γ + κ1 - 2 κ2)^2 + 
   6 s1 (Γ + κ1 - 
      2 κ2) ((Γ + κ1 - 
        2 κ2)^3 + 
      108 g2^2 (-Γ + κ2)) + \
((Γ + κ1 - 2 κ2)^3 + 
     108 g2^2 (-Γ + κ2))^2])^(1/3) + 
2 (Γ + κ1 + κ2 - 6 I ω))

The expression is now simplified in terms of s1. Further observation reveals more simplification. Defining and compiling

j2 = FullSimplify[j1, 
s2 == 4 s1^3 + 
9 s1^2 (Γ + κ1 - 2 κ2)^2 + 
6 s1 (Γ + κ1 - 
   2 κ2) ((Γ + κ1 - 2 κ2)^3 + 
   108 g2^2 (-Γ + κ2)) + ((Γ + \
κ1 - 2 κ2)^3 + 
  108 g2^2 (-Γ + κ2))^2]

now returns a more simplified expression in terms of s1 and s2

1/12 (2 Γ + 2 κ1 - (
2 2^(1/3)
 s1)/(Sqrt[
 s2] - (3 s1 + (Γ + κ1 - 
      2 κ2)^2) (Γ + κ1 - 
    2 κ2) + 108 g2^2 (Γ - κ2))^(
1/3) + 2^(
2/3) (Sqrt[
  s2] - (3 s1 + (Γ + κ1 - 
       2 κ2)^2) (Γ + κ1 - 
     2 κ2) + 108 g2^2 (Γ - κ2))^(
1/3) + 2 κ2 - 12 I ω)

Observing j2 (if you're reading this far, I'd like to sincerely thank you for trying to help), there seems to be one last substitution that can be done. In particular

j3 = FullSimplify[j2, 
t1 == Sqrt[
s2] - (3*
    s1 + (Γ + κ1 - 
     2*κ2)^2)*(Γ + κ1 - 
   2*κ2) + 108*g2^2*(Γ - κ2)]

Should return a simplified expression in terms of s1, s2 and t1. However, the simplification did not happen and j2 looks exactly the same as j3.Why is it that the previous simplification (j1 and j2) works but j3 doesn't?

Lastly, to check if the eigenvalues simplify correctly, doing

valsM3[[1]]-j2//FullSimplify

Should yield 0. But instead, a long expression returns. What is my mistake here?

Thank you for reading this far and I thank you very much for the help in advance.

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