特征值¶

\[ y(k) + a_{n-1} y(k-1) + \cdots + a_0 y(k-n) = 0 \]

\[ \begin{bmatrix} y(k)\\ y(k-1)\\ \vdots\\ y(k-n+1) \end{bmatrix}=\begin{bmatrix} -a_{n-1} &-a_{n-2} &\cdots &-a_0 \\ 1 & & & \\ &\ddots & & \\ & &1 & \end{bmatrix}\begin{bmatrix} y(k-1)\\ y(k-2)\\ \vdots\\ y(k-n) \end{bmatrix} \]

\[ \begin{vmatrix} -a_{n-1}-\lambda &-a_{n-2} &\cdots &-a_0 \\ 1 &-\lambda & & \\ &\ddots &\ddots & \\ & &1 &-\lambda \end{vmatrix}=0 \]

\[ \begin{vmatrix} -\lambda - \displaystyle\sum_{i=0}^{n-1} \lambda^{-i}a_{n-1-i} &-\displaystyle\sum_{i=0}^{n-2} \lambda^{-i}a_{n-2-i} &\cdots &-a_0 \\ &-\lambda & & \\ & &\ddots & \\ & & &-\lambda \end{vmatrix}=0 \]

\[ (-\lambda)^{n-1}\left ( -\lambda - \displaystyle\sum_{i=0}^{n-1} \lambda^{-i}a_{n-1-i} \right )=0 \]

\[ \lambda^n + \displaystyle\sum_{i=0}^{n-1} \lambda^{i}a_{i}=0 \]