grandes-ecoles 2011 QI.B.1

grandes-ecoles · France · centrale-maths2__mp Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties

For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$.
Let $A \in \mathcal{S}_n(\mathbb{R})$. We assume that $A$ is positive definite.
For all $i \in \llbracket 1; n \rrbracket$, show that the matrix $A^{(i)}$ is positive definite and deduce that $\operatorname{det}\left(A^{(i)}\right) > 0$.

For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$.

Let $A \in \mathcal{S}_n(\mathbb{R})$. We assume that $A$ is positive definite.

For all $i \in \llbracket 1; n \rrbracket$, show that the matrix $A^{(i)}$ is positive definite and deduce that $\operatorname{det}\left(A^{(i)}\right) > 0$.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.C QI.D QII.A QII.B QII.C QII.D QII.E QII.F QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.A.5 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 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