proof of Jordan canonical form theorem
This theorem can be proved combining the cyclic decomposition theorem and the primary decomposition theorem. By hypothesis, the characteristic polynomial of factorizes completely over , and then so does the minimal polynomial of (or its annihilator polynomial). This is because the minimal polynomial of has exactly the same factors on as the characteristic polynomial of . Let’s suppose then that the minimal polynomial of factorizes as . We know, by the primary decomposition theorem, that
We know then that has a basis of the form such that each is of the form
So, if we also notice that , we have that in this sub-basis is the Jordan block
So, taking the basis , we have that in this basis has a Jordan form.
This form is unique (except for the order of the blocks) due to the uniqueness of the cyclic decomposition.
|Title||proof of Jordan canonical form theorem|
|Date of creation||2013-03-22 14:15:36|
|Last modified on||2013-03-22 14:15:36|
|Last modified by||CWoo (3771)|