Let be a direct sum of square matrices and . Prove that the minimal polynomial of is the least common multiple of the minimal polynomials of and .
Given a linear transformation on , we let denote the annihilator in of under the action induced by .
Let . If , we have as a linear transformation. Note that . So we have , and thus and . So , where . Conversely, if , then and as linear transformations. Then , so that . So we have .
That is, , where is the minimal polynomial of .
Lemma: In a principal ideal domain , if , then is the least common multiple of and . Proof: Certainly and , so that is a multiple of both and . If is a multiple of and of , then , so that latex d$ is a multiple of .
The result then follows.