grandes-ecoles 2013 QIII.C

grandes-ecoles · France · centrale-maths2__mp Matrices Eigenvalue and Characteristic Polynomial Analysis

For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$ We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.
Determine the eigenvalues of $A_p$.

For $p \in \mathbb{N}^*$, we set
$$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$
We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.

Determine the eigenvalues of $A_p$.

Paper Questions

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