grandes-ecoles 2013 QIII.E.1

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

Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$. Show that there exists a unique triplet $(\alpha, \beta, \gamma)$ of $\mathbb{R}^2 \times \mathbb{R}_+$ that we will specify, such that $A$ is directly orthogonally similar to $\left(\begin{array}{cc} \alpha+\gamma & -\beta \\ \beta & \alpha-\gamma \end{array}\right)$.

Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$. Show that there exists a unique triplet $(\alpha, \beta, \gamma)$ of $\mathbb{R}^2 \times \mathbb{R}_+$ that we will specify, such that $A$ is directly orthogonally similar to $\left(\begin{array}{cc} \alpha+\gamma & -\beta \\ \beta & \alpha-\gamma \end{array}\right)$.

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