grandes-ecoles 2010 QIIC

grandes-ecoles · France · centrale-maths2__pc Matrices Diagonalizability and Similarity

Let $A$ be a non-zero matrix of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$. Show that the following properties are equivalent:
i. The matrix $A$ is nilpotent;
ii. The spectrum of $A$ is equal to $\{ 0 \}$;
iii. The matrix $A$ is similar to the matrix $\left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$.

Let $A$ be a non-zero matrix of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$. Show that the following properties are equivalent:

i. The matrix $A$ is nilpotent;

ii. The spectrum of $A$ is equal to $\{ 0 \}$;

iii. The matrix $A$ is similar to the matrix $\left( \begin{array} { l l } 0 & 1 \\ 0 & 0 \end{array} \right)$.

Paper Questions

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