[lin.alg] [tricks]

$A = \begin{bmatrix}0 & 1 & 0 & \cdots & 0 \\ \vdots & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 1 \\ -a_{n-1} & -a_{n-2} & \cdots & \cdots & -a_{0} \end{bmatrix}$

has characteristic polynomial $z^n + a_{n-1}z^{n-1} + \dotsb + a_0$.

[tricks]
1. For a polynomial $f(z) = c_n z^n + c_{n-1}z^{n-1} + \dotsb + c_0$ with roots $a_1, \dotsc ,\ a_n$,

$\displaystyle \sum a_i = -\frac{c_{n-1}}{c_n},$
$\displaystyle \prod a_i = (-1)^n \frac{c_0}{c_n}$

2. $\displaystyle{f^\prime (z) = f(z) \left( \frac{1}{z-a_1} + \dotsb + \frac{1}{z-a_n} \right)}$