grandes-ecoles 2017 Q9

grandes-ecoles · France · x-ens-maths__pc Matrices Matrix Norm, Convergence, and Inequality
Let $A \in M _ { n } ( \mathbb { C } )$. Show that if $\rho ( A ) < 1$, then the sequence $\left( A ^ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ converges to 0.
Let $A \in M _ { n } ( \mathbb { C } )$. Show that if $\rho ( A ) < 1$, then the sequence $\left( A ^ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ converges to 0.