grandes-ecoles 2023 QExercise-1

grandes-ecoles · France · polytechnique-maths__fui Matrices Diagonalizability and Similarity

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Calculate $J^{n}$ and show that $J$ is diagonalisable.

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by

$$J = \left( \begin{array}{ccccc} 
0 & 1 & 0 & \ldots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots & & \ddots & 0 & 1 \\
1 & 0 & \ldots & 0 & 0
\end{array} \right).$$

Calculate $J^{n}$ and show that $J$ is diagonalisable.

Paper Questions

QExercise-1 QExercise-2 QExercise-3 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20