grandes-ecoles 2017 QI.B.3

grandes-ecoles · France · centrale-maths1__official Matrices Matrix Decomposition and Factorization

We 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 $\operatorname{det}(A) = \operatorname{det}\left(A_{s}\right) \operatorname{det}\left(I_{n} + Q\right)$.
c) Deduce that $\operatorname{det}(A) \geqslant \operatorname{det}\left(A_{s}\right)$.

We 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 $\operatorname{det}(A) = \operatorname{det}\left(A_{s}\right) \operatorname{det}\left(I_{n} + Q\right)$.

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

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