grandes-ecoles 2015 Q13

grandes-ecoles · France · x-ens-maths__psi Matrices Matrix Norm, Convergence, and Inequality

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
Give a necessary and sufficient condition on $m$ for the sequence $\left((I_{3} + \mathcal{M})^{n}\right)_{n \in \mathbb{N}}$ to converge in $\mathcal{M}_{3}(\mathbb{R})$.

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix
$$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$

Give a necessary and sufficient condition on $m$ for the sequence $\left((I_{3} + \mathcal{M})^{n}\right)_{n \in \mathbb{N}}$ to converge in $\mathcal{M}_{3}(\mathbb{R})$.

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