grandes-ecoles 2013 QI.E

grandes-ecoles · France · centrale-maths2__mp Matrices Matrix Group and Subgroup Structure

Let $\varphi$ be the map from $\mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$ to $\mathrm{GL}_n(\mathbb{R})$ defined by $\varphi(O, S) = OS$ for every pair $(O, S)$ of $\mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$.
Show that $\varphi$ is bijective, continuous, and that its inverse is continuous.

Let $\varphi$ be the map from $\mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$ to $\mathrm{GL}_n(\mathbb{R})$ defined by $\varphi(O, S) = OS$ for every pair $(O, S)$ of $\mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$.

Show that $\varphi$ is bijective, continuous, and that its inverse is continuous.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QI.C.3 QI.D.1 QI.D.2 QI.D.3 QI.D.4 QI.E QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QIII.A QIII.B QIII.C QIII.D QIV.A QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.C.1 QIV.C.2 QIV.C.3 QIV.C.4 QIV.C.5