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

Question

    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?

Answer 1

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.