grandes-ecoles 2013 QII.A.3

grandes-ecoles · France · centrale-maths2__mp Matrices Diagonalizability and Similarity

In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a matrix $P \in \mathrm{GL}_n(\mathbb{R})$ such that $A = PBP^{-1}$ and ${}^t A = P {}^t B P^{-1}$. We write $P$ in the form $P = OS$, with $O \in \mathrm{O}(n)$ and $S \in \mathcal{S}_n^{++}(\mathbb{R})$.
a) Show that $BS^2 = S^2 B$, then that $BS = SB$.
b) Deduce that there exists $O \in \mathrm{O}(n)$ such that $A = OB {}^t O$.

In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a matrix $P \in \mathrm{GL}_n(\mathbb{R})$ such that $A = PBP^{-1}$ and ${}^t A = P {}^t B P^{-1}$. We write $P$ in the form $P = OS$, with $O \in \mathrm{O}(n)$ and $S \in \mathcal{S}_n^{++}(\mathbb{R})$.

a) Show that $BS^2 = S^2 B$, then that $BS = SB$.

b) Deduce that there exists $O \in \mathrm{O}(n)$ such that $A = OB {}^t O$.

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