Let be a field, and -dimensional -vector space, and a linear transformation on . Suppose the eigenvalues of are contained in , and let be one such eigenvalue. Let be a natural number, and let . Prove that is the number of Jordan blocks of with eigenvalue and having size .
We will use the notation established in this previous exercise. To wit, are the eigenvalues of , and we have , where is the ‘size’ (that is, dimension over ) of its corresponding Jordan block. For brevity we say .
We showed in that previous exercise that . Thus we have the following.
This final sum is precisely the number of Jordan blocks with eigenvalue and having size , as desired.