# MatrixExp[] for 4 by 4 matrix gives an unavailable answer [duplicate]

    Q = ( {
{-7, 1, 2, 4},
{2, -4, 1, 1},
{2, 2, -6, 2},
{1, 1, 1, -3}
} )
MatrixExp[Q*t]


I defined the matrix Q and executed the command. Then I got a result

{{-(248/137) - 59/274 Root[274 + 130 # + 20 #^2 + #^3& , 1, 0] - (
248 E^(t Root[274 + 130 # + 20 #^2 + #^3& , 1, 0]))/(
137 + 130 Root[274 + 130 # + 20 #^2 + #^3& , 1, 0] +
30 Root[274 + 130 # + 20 #^2 + #^3& , 1, 0]^2 +
2 Root[274 + 130 # + 20 #^2 + #^3& , 1, 0]^3) - (... and so on.)



Is there a way to obtain a cleaner form of the matrix exponential?

• I would argue it is the cleanest form possible. In order to calculate the matrix exponential you need to diagonalize it. You have a matrix 4x4, the diagonalization requires solution of a 4th order algebraic equation, which is not simple. Jun 14 at 11:17
• ToRadicals[MatrixExp[Q*t]] // Simplify
– Acus
Jun 14 at 12:28
• MatrixExp[Q*t] // FullSimplify will give a much shorter form while still using Root expressions Jun 14 at 17:14

The exponential of a matrix is based on finding its eigenvalues. For a $$4\times 4$$ matrix these eigenvalues are the roots of a 4th order polynomial. That's why you see the Root function there. You would get the numerical result by applying N.

If you are looking for an exact solution, first you should determine whether the eigenvalues have any exact form or so. The characteristic polynomial of the matrix is:

Factor@CharacteristicPolynomial[{{-7, 1, 2, 4}, {2, -4, 1, 1}, {2, 2, -6, 2}, {1, 1, 1, -3}}, x]

(** x (274 + 130 x + 20 x^2 + x^3) **)


now you can see that apparently, $$x (x^3+20x^2+ 130 x + 274)$$ has no integer/rational solution other than $$0$$, hence the excessive use of Root function in your answer.