grandes-ecoles 2024 Q1

grandes-ecoles · France · mines-ponts-maths2__mp Matrices Diagonalizability and Similarity
Show that the matrices $M$ and $\left( m _ { \rho ( i ) , \rho ( j ) } \right) _ { 1 \leq i , j \leq n }$ are similar. Deduce that if $G = ( S , A )$ is a non-empty graph, and if $\sigma$ and $\sigma ^ { \prime }$ are two indexings of $S$, then $M _ { G , \sigma }$ and $M _ { G , \sigma ^ { \prime } }$ are similar. 
Show that the matrices $M$ and $\left( m _ { \rho ( i ) , \rho ( j ) } \right) _ { 1 \leq i , j \leq n }$ are similar.\\
Deduce that if $G = ( S , A )$ is a non-empty graph, and if $\sigma$ and $\sigma ^ { \prime }$ are two indexings of $S$, then $M _ { G , \sigma }$ and $M _ { G , \sigma ^ { \prime } }$ are similar.
Paper Questions
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