grandes-ecoles 2013 QIII.E.2

grandes-ecoles · France · centrale-maths2__pc Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix

Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$. According to the values of $(\alpha, \beta, \gamma)$ (where $A$ is directly orthogonally similar to $\left(\begin{array}{cc} \alpha+\gamma & -\beta \\ \beta & \alpha-\gamma \end{array}\right)$), specify the number of real eigenvalues of $A$.

Let $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$. According to the values of $(\alpha, \beta, \gamma)$ (where $A$ is directly orthogonally similar to $\left(\begin{array}{cc} \alpha+\gamma & -\beta \\ \beta & \alpha-\gamma \end{array}\right)$), specify the number of real eigenvalues of $A$.

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