M = {{-I*ω + Γ/2, I*g1,
0}, {I*g1, -I*ω + κ1/2, I*g2}, {0,
I*g2, -I*ω + κ2/2}};
ev = Eigenvalues[M];
The Eigenvalues contain Root
objects. Looking at the first one
ev[[1]]
(* 1/2 Root[-4 g2^2 Γ -
4 g1^2 κ2 - Γ κ1 κ2 +
8 I g1^2 ω + 8 I g2^2 ω +
2 I Γ κ1 ω +
2 I Γ κ2 ω +
2 I κ1 κ2 ω + 4 Γ ω^2 +
4 κ1 ω^2 + 4 κ2 ω^2 - 8 I ω^3 +
4 g1^2 #1 +
4 g2^2 #1 + Γ κ1 #1 + Γ κ2 #1 + \
κ1 κ2 #1 - 4 I Γ ω #1 -
4 I κ1 ω #1 - 4 I κ2 ω #1 -
12 ω^2 #1 - Γ #1^2 - κ1 #1^2 - κ2 #1^2 \
+ 6 I ω #1^2 + #1^3 &, 1] *)
Low-order Root
objects can be converted to radicals with ToRadicals
ev1 = ev[[1]] // ToRadicals
(* 1/2 (-((2^(
1/3) (12 g1^2 +
12 g2^2 - Γ^2 + Γ κ1 - \
κ1^2 + Γ κ2 + κ1 κ2 - \
κ2^2))/(3 (-36 g1^2 Γ + 72 g2^2 Γ +
2 Γ^3 - 36 g1^2 κ1 - 36 g2^2 κ1 -
3 Γ^2 κ1 - 3 Γ κ1^2 +
2 κ1^3 + 72 g1^2 κ2 - 36 g2^2 κ2 -
3 Γ^2 κ2 +
12 Γ κ1 κ2 - 3 κ1^2 κ2 -
3 Γ κ2^2 - 3 κ1 κ2^2 +
2 κ2^3 + √(4 (12 g1^2 +
12 g2^2 - Γ^2 + Γ κ1 - \
κ1^2 + Γ κ2 + κ1 κ2 - \
κ2^2)^3 + (-36 g1^2 Γ + 72 g2^2 Γ +
2 Γ^3 - 36 g1^2 κ1 - 36 g2^2 κ1 -
3 Γ^2 κ1 -
3 Γ κ1^2 + 2 κ1^3 +
72 g1^2 κ2 - 36 g2^2 κ2 -
3 Γ^2 κ2 +
12 Γ κ1 κ2 -
3 κ1^2 κ2 - 3 Γ κ2^2 -
3 κ1 κ2^2 + 2 κ2^3)^2))^(1/3))) + (1/(
3 2^(1/3)))((-36 g1^2 Γ + 72 g2^2 Γ +
2 Γ^3 - 36 g1^2 κ1 - 36 g2^2 κ1 -
3 Γ^2 κ1 - 3 Γ κ1^2 +
2 κ1^3 + 72 g1^2 κ2 - 36 g2^2 κ2 -
3 Γ^2 κ2 + 12 Γ κ1 κ2 -
3 κ1^2 κ2 - 3 Γ κ2^2 -
3 κ1 κ2^2 +
2 κ2^3 + √(4 (12 g1^2 +
12 g2^2 - Γ^2 + Γ κ1 - \
κ1^2 + Γ κ2 + κ1 κ2 - \
κ2^2)^3 + (-36 g1^2 Γ + 72 g2^2 Γ +
2 Γ^3 - 36 g1^2 κ1 - 36 g2^2 κ1 -
3 Γ^2 κ1 - 3 Γ κ1^2 +
2 κ1^3 + 72 g1^2 κ2 - 36 g2^2 κ2 -
3 Γ^2 κ2 +
12 Γ κ1 κ2 - 3 κ1^2 κ2 -
3 Γ κ2^2 - 3 κ1 κ2^2 +
2 κ2^3)^2))^(1/3)) +
1/3 (Γ + κ1 + κ2 - 6 I ω)) *)
The complexity/length of this expression is why a Root
object is used to represent it in more compact form.
To avoid the Root
objects you can use the option Cubics
Eigenvalues[M, Cubics -> True]
EDIT: As a workaround for the Eigenvectors
you can try
Simplify[Eigenvectors[M /. Complex[0, -1] -> mi, Cubics -> True] /. mi -> -I]
Eigenvectors::eivec0
. $\endgroup$