grandes-ecoles 2018 Q35

grandes-ecoles · France · centrale-maths1__official Matrices Matrix Norm, Convergence, and Inequality

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a unit vector $u$ in $\mathbb{R}^{d}$ and define $g(M) = \|M \cdot u\|$. Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$
$$\|M \cdot u\| \leqslant \|M\|_{F}$$

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by

$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$

We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a unit vector $u$ in $\mathbb{R}^{d}$ and define $g(M) = \|M \cdot u\|$. Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$

$$\|M \cdot u\| \leqslant \|M\|_{F}$$

Paper Questions

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