grandes-ecoles 2025 Q16

grandes-ecoles · France · mines-ponts-maths2__mp Matrices Projection and Orthogonality

For every matrix $M \in S_n(\mathbf{R})$ construct a vector subspace $F_M$ of $\mathcal{M}_{n,1}(\mathbf{R})$ of dimension $\pi(M)$ satisfying condition $(\mathcal{C}_M)$: $$\forall X \in F \setminus \{0_{n,1}\} \quad X^\top M X > 0.$$ We thus have $d(M) \geq \pi(M)$.

For every matrix $M \in S_n(\mathbf{R})$ construct a vector subspace $F_M$ of $\mathcal{M}_{n,1}(\mathbf{R})$ of dimension $\pi(M)$ satisfying condition $(\mathcal{C}_M)$:
$$\forall X \in F \setminus \{0_{n,1}\} \quad X^\top M X > 0.$$
We thus have $d(M) \geq \pi(M)$.

Paper Questions

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