grandes-ecoles 2022 Q2

grandes-ecoles · France · mines-ponts-maths2__pc Matrices Matrix Norm, Convergence, and Inequality
If $U \in \mathcal{M}_{n,1}(\mathbf{C})$, show that $$\max\left\{\left|V^{T}U\right|; V \in \Sigma_{n}\right\} = \|U\|.$$ Deduce that, if $M$ is in $\mathcal{M}_{n}(\mathbf{C})$, then $$\max\left\{\left|X^{T}MY\right|; (X,Y) \in \Sigma_{n} \times \Sigma_{n}\right\} = \|M\|_{\mathrm{op}}.$$ 
If $U \in \mathcal{M}_{n,1}(\mathbf{C})$, show that
$$\max\left\{\left|V^{T}U\right|; V \in \Sigma_{n}\right\} = \|U\|.$$
Deduce that, if $M$ is in $\mathcal{M}_{n}(\mathbf{C})$, then
$$\max\left\{\left|X^{T}MY\right|; (X,Y) \in \Sigma_{n} \times \Sigma_{n}\right\} = \|M\|_{\mathrm{op}}.$$
Paper Questions
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