grandes-ecoles 2018 QIII.1

grandes-ecoles · France · x-ens-maths__pc Matrices Matrix Entry and Coefficient Identities

For $A = (a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_{n}(\{-1,1\})$ and $Y = (y_{i})_{1 \leqslant i \leqslant n} \in \{-1,1\}^{n}$, we denote $$g_{A}(Y) = \max\left\{{}^{t}X A Y \mid X \in \{-1,1\}^{n}\right\}.$$
Show that the function $g_{A}$ can be rewritten as $$g_{A}(Y) = \sum_{i=1}^{n} \left|\sum_{j=1}^{n} a_{i,j} y_{j}\right|.$$

For $A = (a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_{n}(\{-1,1\})$ and $Y = (y_{i})_{1 \leqslant i \leqslant n} \in \{-1,1\}^{n}$, we denote
$$g_{A}(Y) = \max\left\{{}^{t}X A Y \mid X \in \{-1,1\}^{n}\right\}.$$

Show that the function $g_{A}$ can be rewritten as
$$g_{A}(Y) = \sum_{i=1}^{n} \left|\sum_{j=1}^{n} a_{i,j} y_{j}\right|.$$

Paper Questions

QI.1 QI.2 QI.3 QI.4 QI.5 QI.6 QII.1 QII.2 QII.3 QII.4 QII.5 QII.6 QIII.1 QIII.2 QIII.3 QIII.4 QIV.1 QIV.2 QIV.3 QIV.4 QV.1 QV.2