“companion matrix”的概念、定义、翻译、参考文献-科学参考

companion matrix

Mathematics

For a monic polynomial f(x) = xⁿ + a_n_-1xⁿ^-1 + … + a₁x + a₀, the companion matrix of f is the n×n matrix

$[\begin{matrix} \begin{matrix} 0 & 0 & \dots \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} & \begin{matrix} \dots & 0 & - a_{0} \\ \dots & 0 & - a_{1} \\ \dots & 0 & - a_{2} \end{matrix} \\ \begin{matrix} ⋮ & 0 & 1 \\ ⋮ & ⋮ & ⋱ \\ 0 & 0 & \dots \end{matrix} & \begin{matrix} ⋱ & ⋮ & ⋮ \\ ⋱ & 0 & - a_{n - 2} \\ 0 & 1 & - a_{n - 1} \end{matrix} \end{matrix}]$

Its characteristic polynomial and minimal polynomial both equal f(x). See rational canonical form.

单词	companion matrix
释义	companion matrix Mathematics For a monic polynomial f(x) = xⁿ + a_n_-1xⁿ^-1 + … + a₁x + a₀, the companion matrix of f is the n×n matrix $[\begin{matrix} \begin{matrix} 0 & 0 & \dots \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} & \begin{matrix} \dots & 0 & - a_{0} \\ \dots & 0 & - a_{1} \\ \dots & 0 & - a_{2} \end{matrix} \\ \begin{matrix} ⋮ & 0 & 1 \\ ⋮ & ⋮ & ⋱ \\ 0 & 0 & \dots \end{matrix} & \begin{matrix} ⋱ & ⋮ & ⋮ \\ ⋱ & 0 & - a_{n - 2} \\ 0 & 1 & - a_{n - 1} \end{matrix} \end{matrix}]$ Its characteristic polynomial and minimal polynomial both equal f(x). See rational canonical form.
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