grandes-ecoles 2014 QVJ

grandes-ecoles · France · centrale-maths1__mp Matrices Determinant and Rank Computation

For $\lambda \in \mathbb{R}$, let $A_\lambda \in \mathcal{M}_d(\mathbb{R})$ be the matrix having only 1's off the diagonal and only $\lambda$ on the diagonal.
Deduce all continuous functions $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$

For $\lambda \in \mathbb{R}$, let $A_\lambda \in \mathcal{M}_d(\mathbb{R})$ be the matrix having only 1's off the diagonal and only $\lambda$ on the diagonal.

Deduce all continuous functions $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying
$$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$

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