Édition de feuilles d'exercices

[planches/ex5219] mines PC 2019 Soit \(A=(a_{i,j})_{1\leqslant i,j\leqslant n}\in\mathscr{S}_n(\mathbf{R})\). On suppose que \(\chi_A=\displaystyle\mathop{\prod}\limits_{k=1}^n(X-a_{k,k})\). Montrer que \(A\) est diagonale.

[concours/ex0926] centrale MP 1997

1. Calculer \(\mathop{\mathchoice{\hbox{inf}}{\hbox{inf}}{\mathrm{inf}}{\mathrm{inf}}}\limits\{\lambda\in\mathbf{R}_+\mid\forall M\in\mathscr{M}_2(\mathbf{R})\quad\mathop{\mathchoice{\hbox{det}}{\hbox{det}}{\mathrm{det}}{\mathrm{det}}}\nolimits M\leqslant\lambda\mathop{\mathchoice{\hbox{tr}}{\hbox{tr}}{\mathrm{tr}}{\mathrm{tr}}}\nolimits({}^tMM)\}\).

2. Calculer \(\mathop{\mathchoice{\hbox{inf}}{\hbox{inf}}{\mathrm{inf}}{\mathrm{inf}}}\limits\{\lambda\in\mathbf{R}_+\mid\forall M\in\mathscr{M}_2(\mathbf{R})\quad\left(\mathop{\mathchoice{\hbox{tr}}{\hbox{tr}}{\mathrm{tr}}{\mathrm{tr}}}\nolimits M\right)^2\leqslant\lambda\mathop{\mathchoice{\hbox{tr}}{\hbox{tr}}{\mathrm{tr}}{\mathrm{tr}}}\nolimits({}^tMM)\}\).

[planches/ex2019] mines MP 2017 Soit \(n\in\mathbf{N}^*\).

1. Déterminer \(\mathop{\mathchoice{\hbox{min}}{\hbox{min}}{\mathrm{min}}{\mathrm{min}}}\limits\{\lambda\in\mathbf{R},\ \forall A\in\mathscr{M}_n(\mathbf{R}),\ (\mathop{\mathchoice{\hbox{tr}}{\hbox{tr}}{\mathrm{tr}}{\mathrm{tr}}}\nolimits A)^2\leqslant\lambda\mathop{\mathchoice{\hbox{tr}}{\hbox{tr}}{\mathrm{tr}}{\mathrm{tr}}}\nolimits{}^tAA\}\).

2. Déterminer \(\mathop{\mathchoice{\hbox{min}}{\hbox{min}}{\mathrm{min}}{\mathrm{min}}}\limits\{\lambda\in\mathbf{R},\ \forall A\in\mathscr{M}_n(\mathbf{R}),\ |\mathop{\mathchoice{\hbox{det}}{\hbox{det}}{\mathrm{det}}{\mathrm{det}}}\nolimits A|^{2/n}\leqslant\lambda\mathop{\mathchoice{\hbox{tr}}{\hbox{tr}}{\mathrm{tr}}{\mathrm{tr}}}\nolimits{}^tAA\}\).

[oraux/ex7899] mines PSI 2013

1. Trouver \(\mathop{\mathchoice{\hbox{inf}}{\hbox{inf}}{\mathrm{inf}}{\mathrm{inf}}}\limits\left\{\vphantom{|_|}\smash{\lambda\in\mathbf{R}_+,\ \forall A\in\mathscr{M}_2(\mathbf{R}),\ (\mathop{\mathchoice{\hbox{tr}}{\hbox{tr}}{\mathrm{tr}}{\mathrm{tr}}}\nolimits A)^2\leqslant\lambda\mathop{\mathchoice{\hbox{tr}}{\hbox{tr}}{\mathrm{tr}}{\mathrm{tr}}}\nolimits({}^tAA)}\right\}\).

2. Trouver \(\mathop{\mathchoice{\hbox{inf}}{\hbox{inf}}{\mathrm{inf}}{\mathrm{inf}}}\limits\left\{\vphantom{|_|}\smash{\lambda\in\mathbf{R}_+,\ \forall A\in\mathscr{M}_2(\mathbf{R}),\ \mathop{\mathchoice{\hbox{det}}{\hbox{det}}{\mathrm{det}}{\mathrm{det}}}\nolimits A\leqslant\lambda\mathop{\mathchoice{\hbox{tr}}{\hbox{tr}}{\mathrm{tr}}{\mathrm{tr}}}\nolimits({}^tAA)}\right\}\).

3. Peut-on généraliser à \(A\in\mathscr{M}_n(\mathbf{R})\) les résultats précédents ?

[planches/ex9682] mines MP 2023 Soit \(n\in\mathbf{N}^*\). Calculer \(\mathop{\mathchoice{\hbox{det}}{\hbox{det}}{\mathrm{det}}{\mathrm{det}}}\nolimits((i\wedge j)_{1\leqslant i,j\leqslant n})\).

Indication : On rappelle que, pour \(N\in\mathbf{N}^*\), \(N=\displaystyle\sum\limits_{d|N}\varphi(d)\) où \(\varphi\) est l’indicatrice d’Euler.