grandes-ecoles 2010 QII.B.3

grandes-ecoles · France · centrale-maths2__mp Matrices Eigenvalue and Characteristic Polynomial Analysis

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We say that $e$ is $q$-orthogonal if and only if, for all $(i,j) \in \{1,\ldots,n\}^2$ with $i \neq j$, $\varphi(e_i, e_j) = 0$.
Suppose that $e$ is simultaneously $q$-orthogonal and $q'$-orthogonal. Show that, for all $i \in \{1,\ldots,n\}$, $e_i$ is an eigenvector of $h^{-1} \circ h'$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.

Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We say that $e$ is $q$-orthogonal if and only if, for all $(i,j) \in \{1,\ldots,n\}^2$ with $i \neq j$, $\varphi(e_i, e_j) = 0$.

Suppose that $e$ is simultaneously $q$-orthogonal and $q'$-orthogonal. Show that, for all $i \in \{1,\ldots,n\}$, $e_i$ is an eigenvector of $h^{-1} \circ h'$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.B.3 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4