[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.

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[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)}