grandes-ecoles 2012 QI.D

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

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
Prove that if $\operatorname { Tr } ( A ) = 0$ then $0 \in R ( A )$.

Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.

Prove that if $\operatorname { Tr } ( A ) = 0$ then $0 \in R ( A )$.

Paper Questions

QI.A QI.B QI.C QI.D QI.E QI.F QII.A QII.B QII.C QII.D QII.E QII.F QII.G QII.H QIII.A QIII.B QIII.C QIII.D QIII.E QIII.F