grandes-ecoles 2013 QV.A.2

grandes-ecoles · France · centrale-maths2__psi Matrices Matrix Power Computation and Application

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 $A$, with the Euclidean division $P_n = Q_n \chi_A + R_n$.
Show that $E(A)$ exists if and only if $\lim_{n \rightarrow \infty} R_n(A)$ exists.

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 $A$, with the Euclidean division $P_n = Q_n \chi_A + R_n$.

Show that $E(A)$ exists if and only if $\lim_{n \rightarrow \infty} R_n(A)$ exists.

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