grandes-ecoles 2023 Q16

grandes-ecoles · France · mines-ponts-maths1__pc Proof Existence Proof

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Show that there exists $\varepsilon_0 > 0$ such that, for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by
$$f_A(t) = \operatorname{det}(A + tM).$$
Show that there exists $\varepsilon_0 > 0$ such that, for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$.

Paper Questions

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