grandes-ecoles 2013 QI.B.2

grandes-ecoles · France · centrale-maths2__pc Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces

Let $A, B \in \mathcal{M}_n(\mathbb{R})$ and $P \in \mathrm{GL}_n(\mathbb{R})$ such that $B = P^{-1}AP$. Show that if $\lambda$ is an eigenvalue of $A$, then $E_\lambda(f_A) = f_P(E_\lambda(f_B))$.

Let $A, B \in \mathcal{M}_n(\mathbb{R})$ and $P \in \mathrm{GL}_n(\mathbb{R})$ such that $B = P^{-1}AP$. Show that if $\lambda$ is an eigenvalue of $A$, then $E_\lambda(f_A) = f_P(E_\lambda(f_B))$.

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