grandes-ecoles 2017 QI.B.2

grandes-ecoles · France · centrale-maths1__official Invariant lines and eigenvalues and vectors Eigenvalue interlacing and spectral inequalities

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

For every real eigenvalue $\lambda$ of $A$, show that $\min \operatorname{sp}_{\mathbb{R}}\left(A_{s}\right) \leqslant \lambda \leqslant \max \operatorname{sp}_{\mathbb{R}}\left(A_{s}\right)$.

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