Let be a field, for some natural number , and a linear transformation on . Make into an -module via as usual. Let be the (distinct) eigenvalues of (suppose these are all in (if not, extend )). By FTFGMPID, we have . We will call each the *size* of its corresponding Jordan block; note that this is precisely its dimension as an -vector space. Letting be fixed (so is a fixed eigenvalue), we say that is the *generalized eigenspace* of with eigenvalue .

Let , and fix an eigenvalue . Show that the nullity of on is the same as the nullity of on , and that these are equal to the number of Jordan blocks of having eigenvalue and size greater than .

(Recall that if is an -linear transformation, then .)

First an editorial note. The mapping is really (a priori) a linear transformation on . However, any transformation which is a polynomial in fixes each Jordan block of , so we can think of it as a transformation on as well. With this interpretation, to show that the nullity of on is the same as the nullity of on , it is enough to show that the kernel of is contained in . To see this, note that if , then by Lemma 3 of this previous exercise, , and thus the kernel of on the Jordan blocks not in is trivial.

Now using Lemmas 2 and 3 from this previous exercise, we have . Note that every -module isomorphism is also and -vector space isomorphism.

Similarly, , and that the mapping realizing this isomorphism is also an -vector space isomorphism.

By Theorem 7 on page 412 of D&F and by the First Isomorphism Theorem, we have . Note that the dimension of over is just .

Thus we have the following, considering as an endomorphism of .

= | ||

= | ||

= | ||

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= | ||

= | . |

Note that this final sum is precisely the number of Jordan blocks with eigenvalue whose size is greater than , as desired.