grandes-ecoles 2013 QIV.C.2

grandes-ecoles · France · centrale-maths2__pc Invariant lines and eigenvalues and vectors Geometric interpretation of eigenstructure

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
Determine the points of $\mathcal{C}(\Omega, r)$ at which the tangent line to $\mathcal{C}(\Omega, r)$ contains $K$.

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.

Determine the points of $\mathcal{C}(\Omega, r)$ at which the tangent line to $\mathcal{C}(\Omega, r)$ contains $K$.

Paper Questions

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