## 每週問題 December 8, 2014

The Leslie matrix is of the form

$\displaystyle L=\begin{bmatrix} a_1&a_2&a_3&\cdots&a_{n-1}&a_n\\ b_1&0&0&\cdots&0&0\\ 0&b_2&0&\cdots&0&0\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\ 0&0&0&\cdots&b_{n-1}&0 \end{bmatrix}$.

Show that the characteristic polynomial of $L$ is

$\displaystyle p(\lambda)=\det(\lambda I-L)=\lambda^n-a_1\lambda^{n-1}-a_2b_1\lambda^{n-2}-a_3b_1b_2\lambda^{n-3}-\cdots-a_nb_1b_2\cdots b_{n-1}$.

\displaystyle \begin{aligned} p(\lambda)&=\begin{vmatrix} \lambda-a_1&-a_2&-a_3&\cdots&-a_{n-1}&-a_n\\ -b_1&\lambda&0&\cdots&0&0\\ 0&-b_2&\lambda&\cdots&0&0\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\ 0&0&0&\cdots&-b_{n-1}&\lambda \end{vmatrix}\\ &=(\lambda-a_1)\begin{vmatrix} \lambda&0&\cdots&0&0\\ -b_2&\lambda&\cdots&0&0\\ \vdots&\vdots&&\vdots&\vdots\\ 0&0&\cdots&-b_{n-1}&\lambda \end{vmatrix}+b_1\begin{vmatrix} -a_2&-a_3&\cdots&-a_{n-1}&-a_n\\ -b_2&\lambda&\cdots&0&0\\ \vdots&\vdots&&\vdots&\vdots\\ 0&0&\cdots&-b_{n-1}&\lambda \end{vmatrix}\\ &=(\lambda-a_1)\lambda^{n-1}-a_2b_1\begin{vmatrix} \lambda&0&\cdots&0&0\\ -b_3&\lambda&\cdots&0&0\\ \vdots&\vdots&&\vdots&\vdots\\ 0&0&\cdots&-b_{n-1}&\lambda \end{vmatrix}+b_1b_2\begin{vmatrix} -a_3&-a_4&\cdots&-a_{n-1}&-a_n\\ -b_3&\lambda&\cdots&0&0\\ \vdots&\vdots&&\vdots&\vdots\\ 0&0&\cdots&-b_{n-1}&\lambda \end{vmatrix}\\ &=\lambda^{n}-a_1\lambda^{n-1}-a_2b_1\lambda^{n-2}+b_1b_2\begin{vmatrix} -a_3&-a_4&\cdots&-a_{n-1}&-a_n\\ -b_3&\lambda&\cdots&0&0\\ \vdots&\vdots&&\vdots&\vdots\\ 0&0&\cdots&-b_{n-1}&\lambda \end{vmatrix}\\ &=\lambda^{n}-a_1\lambda^{n-1}-a_2b_1\lambda^{n-2}-a_3b_1b_2\lambda^{n-3}+b_1b_2b_3\begin{vmatrix} -a_4&-a_5&\cdots&-a_{n-1}&-a_n\\ -b_4&\lambda&\cdots&0&0\\ \vdots&\vdots&&\vdots&\vdots\\ 0&0&\cdots&-b_{n-1}&\lambda \end{vmatrix}\\ &=\cdots\\ &=\lambda^n-a_1\lambda^{n-1}-a_2b_1\lambda^{n-2}-a_3b_1b_2\lambda^{n-3}-\cdots-a_nb_1b_2\cdots b_{n-1}. \end{aligned}

This entry was posted in pow 特徵分析, 每週問題 and tagged , . Bookmark the permalink.

### 1 則回應給 每週問題 December 8, 2014

1. Meiyue Shao 說：

这个问题和Frobenius友阵类似，按照最后一列(column)展开会相对方便一些，正负号不易出错，而且可以直接用归纳法。