Let be a matrix over a field. Prove that is similar to .
We begin with a simple lemma.
Lemma: Let be an invertible matrix. Then is invertible, with . Proof: Recall that . Now .
Let be a square matrix over a field. By Theorem 21 on page 479 in D&F, there exist invertible matrices and such that is in Smith Normal Form. Recall that is diagonal, and that the nonunit diagonal entries are precisely the invariant factors of . Now , and in particular . Now and are invertible, and so is the Smith Normal Form of . In particular, and have the same invariant factors, and thus are similar.