📂Linear Algebra

Direct Sum of Invariant Subspaces and Its Characteristic Polynomial

Theorem¹

Let $T : V \to V$ be a linear transformation on a finite-dimensional vector space $V$ above. Assume that $V$ is the direct sum of the $T$-invariant subspaces $W_{i}$.

$$V = W_{1} \oplus W_{2} \oplus \cdots \oplus W_{k}$$

Let $f_{i}(t)$ be the characteristic polynomial of the restriction $T|_{W_{i}}$. Then, the characteristic polynomial of $T$, $f(t)$, is as follows.

$$f(t) = f_{1}(t) \cdot f_{2}(t) \cdot \cdots \cdot f_{k}(t)$$

Proof

Prove by mathematical induction.

• When $k=2$, it holds.

  Let $\beta_{1}, \beta_{2}$ be the ordered basis of $W_{1}, W_{2}$. And let’s say $\beta = \beta_{1} \cup \beta_{2}$. Then, by the property of direct sums, $\beta$ is the ordered basis of $V$.

  Now, let’s say $A = \begin{bmatrix} T \end{bmatrix}_{\beta}$, $B_{1} = \begin{bmatrix} T_{W_{1}} \end{bmatrix}_{\beta_{1}}$, $B_{2} = \begin{bmatrix} T_{W_{2}} \end{bmatrix}_{\beta_{2}}$. Then, $A$ is represented as the following block matrix. $$A = \begin{bmatrix} B_{1} & O \\ O & B_{2} \end{bmatrix}$$ Here, let $O$ be a zero matrix of appropriate size. Then, by the determinant of block matrices, $$f(t) = \det(A - tI) = \det(B_{1} - tI) \det(B_{2} - tI) = f_{1}(t) \cdot f_{2}(t)$$

• Assuming it holds when $k-1 \ge 2$, it also holds when $k$.

  Let $V$ be the direct sum of the subspaces $W_{i}$. $$V = W_{1} \oplus W_{2} \oplus \cdots \oplus W_{k}$$ Let $W$ be the sum of $W_{i}(1\le i \le k-1)$s. $$W = W_{1} + W_{2} + \cdots + W_{k-1}$$ Then, $W$ is $T$-invariant, and $V = W \oplus W_{k}$ holds. By the proof when $k=2$, if $g(t)$ is the characteristic polynomial of $T|_{W}$, then $f(t) = g(t)f_{k}(t)$. In fact, $W = W_{1} \oplus W_{2} \oplus \cdots \oplus W_{k-1}$ holds and, by assumption, $g(t) = f_{1}(t) \cdots f_{k-1}(t)$. Therefore, $$f(t) = g(t)f_{k}(t) = f_{1}(t) f_{2}(t) \cdots f_{k}(t)$$

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