Given a matrix $A$ with elements $a_{i,j}\in \mathbb C$ I am looking for the eigenvectors.
The question has a background in computing with a computer, and I really despise floating point values with all the messy numerical problems incurred.
Now I wonder if there is a cure for it, given that the $a_{i,j}$ have some restrictions.
(I am thinking of $a_{i,j}\in \mathbb Z$ but would like a more general answer.)
In theory, the thing is easy: take a matrix $A$, solve $\det(A-\lambda I)=0$ and then for each $\lambda$ solve $\vec v (A-\lambda I)=0$.
Inverting a matrix can be done with elementary operations, including division (if the inverse exists).
Complexity is added as the eigenvalues generally are not elements of the minimal field implied by the $a_{i,j}$, but are part of an algebraic extension.
Is there any theory that gives a hint on how to avoid unnecessary complexity?