## 每週問題 April 21, 2014

Let $D$ be the differential operator on $\mathcal{P}_n$ over $\mathbb{R}$ defined as follows: If $p(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in\mathcal{P}_n$, then

$\displaystyle D(p(x))=a_1+2a_2x+\cdots+na_nx^{n-1}$.

(a) Find the matrix representation of $D$ under the basis $\boldsymbol{\beta}=\{1,x,x^2,\ldots,x^n\}$.
(b) Find the eigenvalues of $D$.

(a) 根據線性變換表示矩陣的定義，

\displaystyle\begin{aligned} \begin{bmatrix} D\end{bmatrix}_{\boldsymbol{\beta}}&=\begin{bmatrix} [D(1)]_{\boldsymbol{\beta}}&[D(x)]_{\boldsymbol{\beta}}&[D(x^2)]_{\boldsymbol{\beta}}&\cdots&[D(x^n)]_{\boldsymbol{\beta}} \end{bmatrix}\\ &=\begin{bmatrix} \mathbf{0}&\mathbf{e}_1&2\mathbf{e}_2&\cdots&n\mathbf{e}_n \end{bmatrix}, \end{aligned}

$\displaystyle \begin{bmatrix} D \end{bmatrix}_{\boldsymbol{\beta}}=\begin{bmatrix} 0&1&0&0\\ 0&0&2&0\\ 0&0&0&3\\ 0&0&0&0 \end{bmatrix}$

(b) 因為 $\begin{bmatrix} D \end{bmatrix}_{\boldsymbol{\beta}}$ 是上三角矩陣，主對角元即為其特徵值，故 $D$ 的特徵值全部為零。

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