grandes-ecoles 2011 QI.A

grandes-ecoles · France · centrale-maths2__psi Not Maths

Let $\pi$ be the endomorphism of $\mathbb{R}^n$ whose representation in the canonical basis is the matrix $P = I_n - \frac{1}{n}J$, where $J = Z^t Z$ and $Z = \left(\begin{array}{c}1\\ \vdots \\ 1\end{array}\right) \in \mathcal{M}_{n,1}(\mathbb{R})$.
Show that $\pi$ is an orthogonal projector and specify its characteristic elements.

Let $\pi$ be the endomorphism of $\mathbb{R}^n$ whose representation in the canonical basis is the matrix $P = I_n - \frac{1}{n}J$, where $J = Z^t Z$ and $Z = \left(\begin{array}{c}1\\ \vdots \\ 1\end{array}\right) \in \mathcal{M}_{n,1}(\mathbb{R})$.

Show that $\pi$ is an orthogonal projector and specify its characteristic elements.

Paper Questions

QI.A QI.B QI.C QII.A QII.B QIII.A QIII.B QIV.A.1 QIV.A.2 QIV.A.3 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.B.5 QIV.B.6 QV.A.1 QV.A.2 QV.B.1 QV.B.2 QV.B.3 QV.C.1 QV.C.2 QV.C.3 QV.C.4 QV.C.5 QV.C.6 QV.C.7