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In Mathematica, I computed the following:

Eigenvalues[{{1/3, ((a + Sqrt[3] b ) - I (Sqrt[3] a - b))/6 Sqrt[2], ((a + Sqrt[3] b ) + I (Sqrt[3] a - b))/6 Sqrt[2],1/6}, {((a + Sqrt[3] b ) + I (Sqrt[3] a - b))/6 Sqrt[2], 1/6, ((a + Sqrt[3] b ) - I (Sqrt[3] a - b))/ 6, ((a + Sqrt[3] b ) + I (Sqrt[3] a - b))/6 Sqrt[2]}, {((a + Sqrt[3] b ) - I (Sqrt[3] a - b))/6 Sqrt[2], ((a + Sqrt[3] b ) + I (Sqrt[3] a - b))/6,1/6, ((a + Sqrt[3] b ) - I (Sqrt[3] a - b))/6 Sqrt[2]}, {1/6, ((a + Sqrt[3] b ) - I (Sqrt[3] a - b))/6 Sqrt[2], ((a + Sqrt[3] b ) + I (Sqrt[3] a - b))/6 Sqrt[2], 1/3 }}]

and obtained the answer as

{1/6, 1/6 Root[-3 + 44 a^2 + 64 a^3 + 44 b^2 - 
 192 a b^2 + (7 - 36 a^2 - 36 b^2) #1 - 5 #1^2 + #1^3 &, 1],1/6 Root[-3 + 44 a^2 + 64 a^3 + 44 b^2 - 192 a b^2 + (7 - 36 a^2 - 36 b^2) #1 - 5 #1^2 + #1^3 &, 2], 1/6 Root[-3 + 44 a^2 + 64 a^3 + 44 b^2 - 192 a b^2 + (7 - 36 a^2 - 36 b^2) #1 - 5 #1^2 + #1^3 &, 3]}

Please let me know how to read the last three eigenvalues.

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  • 2
    $\begingroup$ See documentation for Root. $\endgroup$ – Daniel Lichtblau Nov 12 '18 at 15:43
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ev = Eigenvalues[{{1/
     3, ((a + Sqrt[3] b) - I (Sqrt[3] a - b))/6 Sqrt[
      2], ((a + Sqrt[3] b) + I (Sqrt[3] a - b))/6 Sqrt[2], 
    1/6}, {((a + Sqrt[3] b) + I (Sqrt[3] a - b))/6 Sqrt[2], 
    1/6, ((a + Sqrt[3] b) - I (Sqrt[3] a - b))/
     6, ((a + Sqrt[3] b) + I (Sqrt[3] a - b))/6 Sqrt[
      2]}, {((a + Sqrt[3] b) - I (Sqrt[3] a - b))/6 Sqrt[
      2], ((a + Sqrt[3] b) + I (Sqrt[3] a - b))/6, 
    1/6, ((a + Sqrt[3] b) - I (Sqrt[3] a - b))/6 Sqrt[2]}, {1/
     6, ((a + Sqrt[3] b) - I (Sqrt[3] a - b))/6 Sqrt[
      2], ((a + Sqrt[3] b) + I (Sqrt[3] a - b))/6 Sqrt[2], 1/3}}]

(* {1/6, 1/6 Root[-3 + 44 a^2 + 64 a^3 + 44 b^2 - 
     192 a b^2 + (7 - 36 a^2 - 36 b^2) #1 - 5 #1^2 + #1^3 &, 1], 
 1/6 Root[-3 + 44 a^2 + 64 a^3 + 44 b^2 - 
     192 a b^2 + (7 - 36 a^2 - 36 b^2) #1 - 5 #1^2 + #1^3 &, 2], 
 1/6 Root[-3 + 44 a^2 + 64 a^3 + 44 b^2 - 
     192 a b^2 + (7 - 36 a^2 - 36 b^2) #1 - 5 #1^2 + #1^3 &, 3]} *)

ToRadicals will convert the Root objects to (the much more complicated) radical expressions

ev // ToRadicals // Simplify

(* {1/6, (1/72)*(20 + 
        (8*(1 + 27*a^2 + 27*b^2))/
          (1 + 27*a^2 - 108*a^3 + 
               27*b^2 + 324*a*b^2 + 
               Sqrt[-(1 + 27*a^2 + 27*b^2)^
                       3 + (1 + 27*a^2 - 
                        108*a^3 + 27*b^2 + 
                        324*a*b^2)^2])^(1/3) + 
        8*(1 + 27*a^2 - 108*a^3 + 
               27*b^2 + 324*a*b^2 + 
               Sqrt[-(1 + 27*a^2 + 27*b^2)^
                       3 + (1 + 27*a^2 - 
                        108*a^3 + 27*b^2 + 
                        324*a*b^2)^2])^(1/3)), 
   (1/144)*(40 - (8*I*(-I + Sqrt[3])*
             (1 + 27*a^2 + 27*b^2))/
          (1 + 27*a^2 - 108*a^3 + 
               27*b^2 + 324*a*b^2 + 
               Sqrt[-(1 + 27*a^2 + 27*b^2)^
                       3 + (1 + 27*a^2 - 
                        108*a^3 + 27*b^2 + 
                        324*a*b^2)^2])^(1/3) + 
        8*I*(I + Sqrt[3])*
          (1 + 27*a^2 - 108*a^3 + 
               27*b^2 + 324*a*b^2 + 
               Sqrt[-(1 + 27*a^2 + 27*b^2)^
                       3 + (1 + 27*a^2 - 
                        108*a^3 + 27*b^2 + 
                        324*a*b^2)^2])^(1/3)), 
   (1/144)*(40 + (8*I*(I + Sqrt[3])*
             (1 + 27*a^2 + 27*b^2))/
          (1 + 27*a^2 - 108*a^3 + 
               27*b^2 + 324*a*b^2 + 
               Sqrt[-(1 + 27*a^2 + 27*b^2)^
                       3 + (1 + 27*a^2 - 
                        108*a^3 + 27*b^2 + 
                        324*a*b^2)^2])^(1/3) - 
        8*(1 + I*Sqrt[3])*
          (1 + 27*a^2 - 108*a^3 + 
               27*b^2 + 324*a*b^2 + 
               Sqrt[-(1 + 27*a^2 + 27*b^2)^
                       3 + (1 + 27*a^2 - 
                        108*a^3 + 27*b^2 + 
                        324*a*b^2)^2])^(1/3))} *)

EDIT: Alternatively, use the option Cubics in Eigenvalues

ev = Eigenvalues[{{1/
     3, ((a + Sqrt[3] b) - I (Sqrt[3] a - b))/6 Sqrt[
      2], ((a + Sqrt[3] b) + I (Sqrt[3] a - b))/6 Sqrt[2], 
    1/6}, {((a + Sqrt[3] b) + I (Sqrt[3] a - b))/6 Sqrt[2], 
    1/6, ((a + Sqrt[3] b) - I (Sqrt[3] a - b))/
     6, ((a + Sqrt[3] b) + I (Sqrt[3] a - b))/6 Sqrt[
      2]}, {((a + Sqrt[3] b) - I (Sqrt[3] a - b))/6 Sqrt[
      2], ((a + Sqrt[3] b) + I (Sqrt[3] a - b))/6, 
    1/6, ((a + Sqrt[3] b) - I (Sqrt[3] a - b))/6 Sqrt[2]}, {1/
     6, ((a + Sqrt[3] b) - I (Sqrt[3] a - b))/6 Sqrt[
      2], ((a + Sqrt[3] b) + I (Sqrt[3] a - b))/6 Sqrt[2], 1/3}}, 
  Cubics -> True]//Simplify
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