grandes-ecoles 2014 QIIIB1

grandes-ecoles · France · centrale-maths1__mp Matrices Linear System and Inverse Existence

We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$.
For $R$ large enough, show that, for every $\theta \in \mathbb{R}$, the matrix $(R\mathrm{e}^{\mathrm{i}\theta} I_d - A)$ is invertible in $\mathcal{M}_d(\mathbb{C})$, and that its inverse is the matrix $$\left(R\mathrm{e}^{\mathrm{i}\theta}\right)^{-1} \sum_{n=0}^{+\infty} \left(R\mathrm{e}^{\mathrm{i}\theta}\right)^{-n} A^n$$

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

For $R$ large enough, show that, for every $\theta \in \mathbb{R}$, the matrix $(R\mathrm{e}^{\mathrm{i}\theta} I_d - A)$ is invertible in $\mathcal{M}_d(\mathbb{C})$, and that its inverse is the matrix
$$\left(R\mathrm{e}^{\mathrm{i}\theta}\right)^{-1} \sum_{n=0}^{+\infty} \left(R\mathrm{e}^{\mathrm{i}\theta}\right)^{-n} A^n$$

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