grandes-ecoles 2014 QIIIB2

grandes-ecoles · France · centrale-maths1__mp Matrices Matrix Power Computation and Application

We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$.
Show that, for every $n \in \mathbb{N}^*$ and every $R$ large enough, the matrix $$\frac{1}{2\pi} \int_0^{2\pi} \left(R\mathrm{e}^{\mathrm{i}\theta}\right)^n \left(R\mathrm{e}^{\mathrm{i}\theta} I_d - A\right)^{-1} \mathrm{~d}\theta$$ equals $A^{n-1}$.

We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$.

Show that, for every $n \in \mathbb{N}^*$ and every $R$ large enough, the matrix
$$\frac{1}{2\pi} \int_0^{2\pi} \left(R\mathrm{e}^{\mathrm{i}\theta}\right)^n \left(R\mathrm{e}^{\mathrm{i}\theta} I_d - A\right)^{-1} \mathrm{~d}\theta$$
equals $A^{n-1}$.

Paper Questions

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