grandes-ecoles 2017 QI.B.2

grandes-ecoles · France · centrale-maths1__mp Matrices Eigenvalue and Characteristic Polynomial Analysis

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$. For every real eigenvalue $\lambda$ of $A$, show that $\min \operatorname{sp}_{\mathbb{R}}(A_{s}) \leqslant \lambda \leqslant \max \operatorname{sp}_{\mathbb{R}}(A_{s})$.
Deduce that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$ then $A$ is invertible.

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$. For every real eigenvalue $\lambda$ of $A$, show that $\min \operatorname{sp}_{\mathbb{R}}(A_{s}) \leqslant \lambda \leqslant \max \operatorname{sp}_{\mathbb{R}}(A_{s})$.

Deduce that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$ then $A$ is invertible.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QI.C.1 QI.C.2 QI.C.3 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.A.6 QII.A.7 QII.A.8 QII.B.1 QII.B.2 QII.B.3 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QII.C.6 QII.C.7 QII.C.8 QII.D.1 QII.D.2 QII.E.1 QII.E.2 QII.E.3 QII.E.4 QII.E.5 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3