Ivan Shishkin, Birch Grove

Polynôme annulateur

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Définition formelle

Soit AA une matrice carrée sur un corps K\mathbb{K}.
Un polynôme annulateur de AA est un polynôme P∈K[X]P\in \mathbb{K}[X] tel que
P(A)=0.P(A)=0.

Exemple
  • Par exemple, pour A=(100−1)A=\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}, le polynôme
    P(X)=X2−1P(X)=X^2-1est un polynôme annulateur de AA, car
    P(A)=A2−I2=(1001)−(1001)=0.P(A)=A^2-I_2 = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} - \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} = 0.
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