# Eigenvalues for a $4 \times 4$ matrix [duplicate]

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 }}]


{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]}


• See documentation for Root. – Daniel Lichtblau Nov 12 '18 at 15:43

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