grandes-ecoles 2019 Q19

grandes-ecoles · France · centrale-maths2__psi Matrices Linear System and Inverse Existence

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$. Assume that $P$ is an annihilating polynomial of $A$ nilpotent.
We denote by $m$ the multiplicity of 0 in $P$, which allows us to write $P = X^m Q$ where $Q$ is a polynomial in $\mathbb{C}[X]$ such that $Q(0) \neq 0$. Prove that $Q(A)$ is invertible and then that $P$ is a multiple of $X^p$ in $\mathbb{C}[X]$.

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$. Assume that $P$ is an annihilating polynomial of $A$ nilpotent.

We denote by $m$ the multiplicity of 0 in $P$, which allows us to write $P = X^m Q$ where $Q$ is a polynomial in $\mathbb{C}[X]$ such that $Q(0) \neq 0$. Prove that $Q(A)$ is invertible and then that $P$ is a multiple of $X^p$ in $\mathbb{C}[X]$.

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