The Invariant Subspace
Definition 1. Let . Then is called a -invariant subspace of if .
의 image가 다시 에 포함될 때 를 -불변 부분공간이라고 부른다. 자명하게 는 -불변 부분공간임을 알 수 있다.
The restriction of a Linear Operator
Definition 2. Let , and let be a -invariant subspace of . Then the restriction of to is an linear operator .
정의역을 로 축소시킨 의 restriction, 즉 축소사상은 공역 또한 축소시킨다.
Theorem 1
Theorem 1. Let , and let be a -invariant subspace of (V is finite-dimensional). Then the characteristic polynomial of divides the characteristic polynomial of .
Proof. Let be an ordered basis for . Extend it to an ordered basis for . Let and let . Note that .
The characteristic polynomial of is . Then the characteristic polynomial of is = = = . Thus divides .
즉 어떤 불변 부분공간의 축소사상의 특성다항식을 계산하면 원 사상의 특성다항식의 정보를 알 수 있다. 이를 확장해서, 만일 가 불변 부분공간들의 직합이라면 어떤 사상의 특성다항식은 축소사상들의 특성다항식으로 분해된다.
Theorem 2
Theorem 2. Let , and suppose that , where is a -invariant subspace of for each ( is finite-dimensional). Suppose that is the characteristic polynomial of . Then is the characteristic polynomial of .
Proof. By Theorem 1, there exists an ordered basis for such that is an ordered basis for . Then denote and for . Then by Theorem 2, . Thus the characteristic polynomial of is .