grandes-ecoles 2013 QV.A.1

grandes-ecoles · France · centrale-maths2__psi Matrices Eigenvalue and Characteristic Polynomial Analysis

Let $A \in M_p(\mathbb{C})$ and $n \in \mathbb{N}^*$. We denote $$P_n(X) = \left(1 + \frac{X}{n}\right)^n \in \mathbb{C}[X]$$ and $\chi_A$ the characteristic polynomial of $A$ defined by $$\chi_A(X) = \det\left(A - XI_p\right)$$
Show that there exists a unique pair $\left(Q_n, R_n\right) \in \mathbb{C}[X] \times \mathbb{C}_{p-1}[X]$ such that $$P_n = Q_n \chi_A + R_n$$

Let $A \in M_p(\mathbb{C})$ and $n \in \mathbb{N}^*$. We denote
$$P_n(X) = \left(1 + \frac{X}{n}\right)^n \in \mathbb{C}[X]$$
and $\chi_A$ the characteristic polynomial of $A$ defined by
$$\chi_A(X) = \det\left(A - XI_p\right)$$

Show that there exists a unique pair $\left(Q_n, R_n\right) \in \mathbb{C}[X] \times \mathbb{C}_{p-1}[X]$ such that
$$P_n = Q_n \chi_A + R_n$$

Paper Questions

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