The Invariant Subspace

The Invariant Subspace

Definition 1. Let $T \in \mathcal{L}(V)$. Then $W \leq V$ is called a $T$-invariant subspace of $V$ if $T(W) \subseteq W$.

    $W$의 image가 다시 $W$에 포함될 때 $W$를 $T$-불변 부분공간이라고 부른다. 자명하게 $\{\mathbf{0}\}, V, R(T), N(T), E_{\lambda}$는 $T$-불변 부분공간임을 알 수 있다.

The restriction of a Linear Operator

Definition 2. Let $T \in \mathcal{L}(V)$, and let $W$ be a $T$-invariant subspace of $V$. Then the restriction $T_W$ of $T$ to $W$ is an linear operator $T_W \in \mathcal{L}(W)$.

    정의역을 $W$로 축소시킨 $T$의 restriction, 즉 축소사상은 공역 또한 축소시킨다. 

Theorem 1

Theorem 1. Let $T \in \mathcal{L}(V)$, and let $W$ be a $T$-invariant subspace of $V$ (V is finite-dimensional). Then the characteristic polynomial of $T_W$ divides the characteristic polynomial of $T$.

Proof. Let $\beta = \{v_1, ..., v_p\}$ be an ordered basis for $W$. Extend it to an ordered basis $\gamma = \{v_1, ..., v_p, v_{p+1}, ..., v_n\}$ for $V$. Let $A = [T_W]_{\beta}$ and let $B = [T]_{\gamma}$. Note that $B = \begin{pmatrix} A & C \\ O & D \end{pmatrix}$.
The characteristic polynomial $g(t)$ of $T_W$ is $g(t) = \det(A - tI_p)$. Then the characteristic polynomial $f(t)$ of $T$ is $f(t) = \det(B - tI_n)$ = $\det \begin{pmatrix} A - tI_p & C \\ O & D-tI_{n-p} \end{pmatrix}$ = $\det(A - tI_p) \det(D - tI_{n-p})$ = $g(t) \det(D - tI_{n - p})$. Thus $g(t)$ divides $f(t)$. $\blacksquare$

즉 어떤 불변 부분공간의 축소사상의 특성다항식을 계산하면 원 사상의 특성다항식의 정보를 알 수 있다. 이를 확장해서, 만일 $V$가 불변 부분공간들의 직합이라면 어떤 사상의 특성다항식은 축소사상들의 특성다항식으로 분해된다.

Theorem 2

Theorem 2. Let $T \in \mathcal{L}(V)$, and suppose that $V = \bigoplus_{i=1}^k W_i$, where $W_i$ is a $T$-invariant subspace of $V$ for each $i (1 \leq i \leq k)$ ($V$ is finite-dimensional). Suppose that $f_i(t)$ is the characteristic polynomial of $T_{W_i} (1 \leq i \leq k)$. Then $f_1(t) \cdots f_k(t)$ is the characteristic polynomial of $T$.

Proof. By Theorem 1, there exists an ordered basis $\beta_i$ for $W_i (1 \leq i \leq k)$ such that $\beta = \cup_{i=1}^k \beta_i$ is an ordered basis for $V$. Then denote $A = [T]_{\beta}$ and $B_i = [T_{W_i}]_{\beta_i}$ for $1 \leq i \leq k$. Then by Theorem 2, $A = \bigoplus_{i=1}^k B_i$. Thus the characteristic polynomial $f(t)$ of $T$ is $f(t) = \det(A - tI) = \det(B_1 - tI) \cdots \det(B_k - tI) = f_1(t) \cdots f_k(t)$. $\blacksquare$