grandes-ecoles 2013 QIV.C.3

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

In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $A$ is the matrix such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$, with $A^{-1}$ as given in IV.C.2.
Determine the eigenvalues of $A^{-1} {}^t A^{-1}$.

In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $A$ is the matrix such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$, with $A^{-1}$ as given in IV.C.2.

Determine the eigenvalues of $A^{-1} {}^t A^{-1}$.

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