grandes-ecoles 2017 QI.B.3

grandes-ecoles · France · centrale-maths1__mp Matrices Matrix Decomposition and Factorization

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$ and assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
a) Show that there exists a unique matrix $B$ in $\mathcal{S}_{n}^{++}(\mathbb{R})$ such that $B^{2} = A_{s}$.
b) Show that there exists a matrix $Q$ in $\mathcal{A}_{n}(\mathbb{R})$ such that $\operatorname{det}(A) = \operatorname{det}(A_{s})\operatorname{det}(I_{n} + Q)$.
c) Deduce that $\operatorname{det}(A) \geqslant \operatorname{det}(A_{s})$.

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$ and assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$.

a) Show that there exists a unique matrix $B$ in $\mathcal{S}_{n}^{++}(\mathbb{R})$ such that $B^{2} = A_{s}$.

b) Show that there exists a matrix $Q$ in $\mathcal{A}_{n}(\mathbb{R})$ such that $\operatorname{det}(A) = \operatorname{det}(A_{s})\operatorname{det}(I_{n} + Q)$.

c) Deduce that $\operatorname{det}(A) \geqslant \operatorname{det}(A_{s})$.

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