Let be an matrix over a field . Show that if , (all contained in ), are the eigenvalues of , then are the eigenvalues of .
We begin with a lemma.
Lemma: If and are upper triangular matrices with diagonal entries all equal to and , respectively, then is upper triangular with diagonal entries all equal to . Proof: Certainly is upper triangular. (We showed this here.) Recall now that if , then we have if and if . Similarly if , then if and if . Now . Consider this sum with . If , then . If , then . If , then . In particular, the -entry of is as desired.
Note that if is a Jordan block whose diagonal entries are all , then using the lemma and induction we see that is upper triangular with diagonal entries all equal to .
Now suppose is an arbitrary matrix with eigenvalues . By Theorem 23, is similar to a matrix in Jordan canonical form. In particular, and have the same eigenvalues. If , then , so that and are similar, and thus have the same eigenvalues. Now where are Jordan blocks, so that . The eigenvalues of (diagonal entries) are thus , so that the eigenvalues of are as desired.