grandes-ecoles 2017 QII.A.8

grandes-ecoles · France · centrale-maths1__official Matrices Projection and Orthogonality
We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. We assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Show that $A$ is $H$-regular for every hyperplane $H$ of $E_{n}$. 
We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. We assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Show that $A$ is $H$-regular for every hyperplane $H$ of $E_{n}$.
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