Unable to evaluate Eigenvalues and Eigenvectors for a matrix

Question

I have the following 3X3 matrix M and I wish to find its eigenvectors and eigenvalues

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

Simply doing

Eigenvectors[M]

Returns the error

Eigenvectors: Unable to find all eigenvectors.

Doing

Eigenvalues[M] 

Returns a long list of numbers, variables and the complex identity I. Also some numbers or variables has a # sign attached to it. I suspect it's a condition statement but I'm not sure how to deal with it. Would appreciate any assistance I can get. Thanks for the help in advance.

Answer 1

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]

Answer 2

I tried this and works:

 M = {{-I*ω + Γ/2, I*g1, 
        0}, {I*g1, -I*ω + κ1/2, I*g2}, {0, 
        I*g2, -I*ω + κ2/2}};
    EVec = Eigenvectors[M]; (*gives all eigen vectors*)
    EVec[[1]] (*gives first eigen vector*)
    Eval = Eigenvalues[M]; (*gives all eigen values*)
    Eval[[1]]  (*gives first eigen value*)

You may try this as well,

 {eval, evec} = Eigensystem[M];
    eval 
    evec