# eigenvalues and eigenvectors without Root answers

I'd like to find eigen values and vectors of parametric 21*21 Matrix and i cant solve the root problem.please help me to find the answers without Root. My Matrix is:

 {{-100 d, 0, 2 Sqrt[190] e, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

0, 0, 0, 0}, {0, -81 d, 0, 6 Sqrt[57] e, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0}, {2 Sqrt[190] e, 0, -64 d, 0, 6 Sqrt[102] e,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 6 Sqrt[57] e,
0, -49 d, 0, 8 Sqrt[85] e, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0}, {0, 0, 6 Sqrt[102] e, 0, -36 d, 0, 60 Sqrt[2] e, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 8 Sqrt[85] e, 0, -25 d, 0,
42 Sqrt[5] e, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0,
60 Sqrt[2] e, 0, -16 d, 0, 28 Sqrt[13] e, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0}, {0, 0, 0, 0, 0, 42 Sqrt[5] e, 0, -9 d, 0, 104 e, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 28 Sqrt[13] e, 0, -4 d,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0,
12 Sqrt[78] e, 0, -d, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0,
0, 0, 0, 0, 6 Sqrt[330] e, 0, 0, 0, 6 Sqrt[330] e, 0, 0, 0, 0, 0,
0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -d, 0, 12 Sqrt[78] e, 0,
0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4 d, 0,
28 Sqrt[13] e, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
12 Sqrt[78] e, 0, -9 d, 0, 42 Sqrt[5] e, 0, 0, 0, 0, 0}, {0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 28 Sqrt[13] e, 0, -16 d, 0, 60 Sqrt[2] e,
0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42 Sqrt[5] e,
0, -25 d, 0, 8 Sqrt[85] e, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 60 Sqrt[2] e, 0, -36 d, 0, 6 Sqrt[102] e, 0, 0}, {0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8 Sqrt[85] e, 0, -49 d, 0,
6 Sqrt[57] e, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
6 Sqrt[102] e, 0, -64 d, 0, 2 Sqrt[190] e}, {0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 6 Sqrt[57] e, 0, -81 d, 0}, {0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2 Sqrt[190] e,
0, -100 d}}

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Hello ! Um, what is the root problem ? – Sektor Feb 13 '14 at 18:16
Try setting Cubics->False and Quartics->False. If that does not do the job, could post-process with ToRadicals and maybe that will remove the Root objects from the result. – Daniel Lichtblau Feb 13 '14 at 19:41
You are most likely aware of this, but just in case ... the roots of polynomials of degree 5 or higher can not always be expressed in terms of radicals. What you are asking for may not be possible. There are many symbolic operations that can be done on roots of polynomials though, which is why Mathematica has a symbolic representation for them in the form of Root objects. For example, you can take one the Root objects you get as an answer, take the derivative, and get a result in terms of other Root objects. – Szabolcs Feb 13 '14 at 20:19
Generally, it is not always a necessary (and it's not always a good idea!) to try to get solutions in closed, explicit form. Keep in mind your final goal and think about whether you really need an "explicit" expression. – Szabolcs Feb 13 '14 at 20:20
Using version 9.0.1 for OS X, I don't see any Root objects in {evals, evects} = <your matrix> // Eigensystem // Simplify. Mathematica symbolically solves your eigensystem problem with no fuss. – Stephen Luttrell Feb 14 '14 at 14:52