> Given that we talk about > A = matrix([[0,-1,0,0],[1,0,0,0],[0,0,0,-1],[0,0,1,0]]) # no field > explicitly specified > do you suggest that Sage should restrict itself to eigenvalues in Z, > which is the base ring of A? > Do you suggest that Sage should check whether we create a proper > extension of a base ring when we adjoin > a particular eigenvalue to it? > I am not suggesting anything. What i am saying is that the question may be more complicated than just a "take all the complex roots and that's the only possible choice; period".
In your example, i don't see any reason why we should look for the roots over the complexes and not over the algebraic closure of a finite field, for example. > Eigenvalues, without a specialization to a ring/field, are the roots > of characteristic polynomial, that's it. Eigenvectors are, by definition, the elements of a vector field whose image under a certain endomorpshism is a multiple of them. And eigenvalues are, by definition, the scalars that multiply the eigenvectors. So no, it is not true that "eigenvalues are the roots of a charcteristic polynomial and thats it". Actually, the concept of eigenvalue does not make sense without specializing to a base ring. We may discuss what should be the default specialization when there is none given, but that would be just the chosen convention, not "the only possible right answer". -- To post to this group, send an email to [email protected] To unsubscribe from this group, send an email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
