grandes-ecoles 2013 QIV.B.4

grandes-ecoles · France · centrale-maths2__mp Matrices Projection and Orthogonality

Let $f$ be a linear form on $\mathcal{M}_n(\mathbb{R})$, and let $A$ be the unique matrix in $\mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$. Let $\mu_1, \ldots, \mu_n$ be the $n$ positive eigenvalues of ${}^t A A$ counted with multiplicities, and $D$ the diagonal matrix whose diagonal elements are $\sqrt{\mu_1}, \ldots, \sqrt{\mu_n}$.
Deduce that $M_n = \sum_{k=1}^n \sqrt{\mu_k}$.

Let $f$ be a linear form on $\mathcal{M}_n(\mathbb{R})$, and let $A$ be the unique matrix in $\mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$. Let $\mu_1, \ldots, \mu_n$ be the $n$ positive eigenvalues of ${}^t A A$ counted with multiplicities, and $D$ the diagonal matrix whose diagonal elements are $\sqrt{\mu_1}, \ldots, \sqrt{\mu_n}$.

Deduce that $M_n = \sum_{k=1}^n \sqrt{\mu_k}$.

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