Talk:Minimal polynomial (linear algebra)

Untitled

Could someone finish the example or give a better one? I was trying to follow, but it seems to say "here is an example which as you can see is easy". I would really appreciate a good finished example or a good link for how to solve for it.

Agreed. Just saw this article for the first time, and the example (along with a lot of the article) should definitely be expanded. I think the "Computation" section would also benefit from descriptions of the other common methods by which minimal polynomials are computed beginning with various pieces of information ('e.g.', a description of the techniques one can use to compute the minimal polynomial of some operator given (or after determining) its characteristic polynomial, eigenvalues, etc.). If no one beats me to it, I'll start expanding this article once I have some time to kill. --mathemajor (talk) 00:11, 23 March 2011 (UTC)

I guess I did beat you to it. I hope the text explains that it is perfectly unnecessary to know the characteristic polynomial or the eigenvalues to compute the minimal polynomial; all you need is Gaussian elimination (and the given example is so simple that I just solved the system by inspection). cases where the minimal polynomial for one vector is not already the entire minimal polynomial are rare, and if they do occur, so much the better because one gets a partial factorization of the minimal polynomial. One might pose a more philosophical question of why computing the minimal polynomial is not often used in education as the preferred method to solve concrete eigenvalue problems, rather that the characteristic polynomial: it is not in principle any harder to compute (and easier to teach as no determinants are involved), it could be of a lower degree, and its factorization gives immediate information about diagonalizability, which the characteristic polynomial does not. The argument that shows that the roots of the minimal polynomial give all eigenvalues is slightly more abstract than for the characteristic polynomial, but quite simple (and even talking about the characteristic polynomial properly requires considering determinants of matrices with entries in a polynomial ring, which is not without abstraction either). Maybe the coefficients in the computation of the minimal polynomial get nasty somewhat more easily than for the characteristic polynomial, but determining the characteristic polynomial for a 4×4 matrix without zeros in not a piece of cake either. Marc van Leeuwen (talk) 09:11, 2 April 2011 (UTC)

i think the first equivalence stated is not quire correct. the problem is: if one eigenvalue is zero the corresponding term need not occur in the minimal polynomial. 131.159.0.95 (talk) 10:50, 20 December 2012 (UTC)