grandes-ecoles 2023 Q15

grandes-ecoles · France · mines-ponts-maths1__psi Differentiating Transcendental Functions Regularity and smoothness of transcendental functions

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that $f _ { A }$ is of class $C ^ { \infty }$ on $\mathbf { R }$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by

$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$

Show that $f _ { A }$ is of class $C ^ { \infty }$ on $\mathbf { R }$.

Paper Questions

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