grandes-ecoles 2013 QII.B.3

grandes-ecoles · France · centrale-maths2__psi Matrices Matrix Norm, Convergence, and Inequality

Let $B \in M_3(\mathbb{R})$ be antisymmetric. Assume that $$B = P \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -\beta \\ 0 & \beta & 0 \end{pmatrix} P^{-1}$$ for some $P \in O_3(\mathbb{R})$ and $\beta \in \mathbb{R}$.
Show that $|\beta| = \frac{\|B\|_2}{\sqrt{2}}$.

Let $B \in M_3(\mathbb{R})$ be antisymmetric. Assume that
$$B = P \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -\beta \\ 0 & \beta & 0 \end{pmatrix} P^{-1}$$
for some $P \in O_3(\mathbb{R})$ and $\beta \in \mathbb{R}$.

Show that $|\beta| = \frac{\|B\|_2}{\sqrt{2}}$.

Paper Questions

QI.A QI.B QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIV.A.1 QIV.A.2 QIV.B QIV.C QIV.D QIV.E QIV.F QV.A.1 QV.A.2 QV.A.3 QV.A.4 QV.B.1 QV.B.2 QV.B.3 QV.B.4