grandes-ecoles 2018 Q28

grandes-ecoles · France · centrale-maths1__psi Matrices Linear Transformation and Endomorphism Properties

Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. Let $g \in \mathcal{C}(f_M)$. Show that there exist $(\alpha_0, \ldots, \alpha_{n-1}) \in \mathbb{C}^n$ such that $g = \alpha_0 Id_{\mathbb{C}^n} + \alpha_1 f_M + \cdots + \alpha_{n-1} f_M^{n-1}$. One may use the basis $(x_0, f_M(x_0), \ldots, f_M^{n-1}(x_0))$ and express $g(x_0)$ in this basis.

Let $M$ be a cyclic matrix and $x_0$ be a cyclic vector of $f_M$. Let $g \in \mathcal{C}(f_M)$. Show that there exist $(\alpha_0, \ldots, \alpha_{n-1}) \in \mathbb{C}^n$ such that $g = \alpha_0 Id_{\mathbb{C}^n} + \alpha_1 f_M + \cdots + \alpha_{n-1} f_M^{n-1}$. One may use the basis $(x_0, f_M(x_0), \ldots, f_M^{n-1}(x_0))$ and express $g(x_0)$ in this basis.

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