grandes-ecoles 2016 QI.A.2

grandes-ecoles · France · centrale-maths1__mp 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning

For every matrix $A$ in $\mathcal{M}_n(\mathbb{K})$, we set $N(A) = \max_{1 \leqslant i \leqslant n}\left(\sum_{j=1}^{n}|a_{i,j}|\right)$. Let $Q \in \mathrm{GL}_n(\mathbb{K})$. Show that $A \mapsto \|A\| = N\left(Q^{-1}AQ\right)$ is a sub-multiplicative norm on $\mathcal{M}_n(\mathbb{K})$.

For every matrix $A$ in $\mathcal{M}_n(\mathbb{K})$, we set $N(A) = \max_{1 \leqslant i \leqslant n}\left(\sum_{j=1}^{n}|a_{i,j}|\right)$. Let $Q \in \mathrm{GL}_n(\mathbb{K})$. Show that $A \mapsto \|A\| = N\left(Q^{-1}AQ\right)$ is a sub-multiplicative norm on $\mathcal{M}_n(\mathbb{K})$.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QII.A QII.B QII.C QIII.A QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C QV.A.1 QV.A.2 QV.A.3 QV.A.4 QV.A.5 QV.B.1 QV.B.2 QV.C.1 QV.C.2 QVI.A QVI.B.1 QVI.B.2 QVI.B.3 QVI.C.1 QVI.C.2 QVI.D