grandes-ecoles 2017 QI.A.2

grandes-ecoles · France · centrale-maths1__official Matrices Projection and Orthogonality

Let $A \in \mathcal{M}_{n}(\mathbb{R})$. Show that for every matrix $S \in \mathcal{S}_{n}(\mathbb{R}), \left\|A - A_{s}\right\|_{2} \leqslant \|A - S\|_{2}$. Specify under what condition on $S \in \mathcal{S}_{n}(\mathbb{R})$ this inequality is an equality.

Let $A \in \mathcal{M}_{n}(\mathbb{R})$. Show that for every matrix $S \in \mathcal{S}_{n}(\mathbb{R}), \left\|A - A_{s}\right\|_{2} \leqslant \|A - S\|_{2}$. Specify under what condition on $S \in \mathcal{S}_{n}(\mathbb{R})$ this inequality is an equality.

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