grandes-ecoles 2011 QV.B.1

grandes-ecoles · France · centrale-maths2__psi Invariant lines and eigenvalues and vectors Invariant subspaces and stable subspace analysis

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.
Let $A \in \mathcal{M}_n(\mathbb{R})$. Show that the hyperplane $\mathcal{H}$ with normal vector $Z$ (and equation $x_1 + \cdots + x_n = 0$) is stable under the canonical endomorphism associated with the matrix $\Psi(A)$.

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.

Let $A \in \mathcal{M}_n(\mathbb{R})$. Show that the hyperplane $\mathcal{H}$ with normal vector $Z$ (and equation $x_1 + \cdots + x_n = 0$) is stable under the canonical endomorphism associated with the matrix $\Psi(A)$.

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