Let $T$ be an upper triangular matrix, $A = N + T$ and $k \geqslant 0$. Show that $A$ is similar to a matrix $L$ whose diagonal coefficients of order $k$ are all equal and satisfying $\forall i \in \llbracket -1, k-1 \rrbracket, L^{(i)} = A^{(i)}$.
Let $T$ be an upper triangular matrix, $A = N + T$ and $k \geqslant 0$. Show that $A$ is similar to a matrix $L$ whose diagonal coefficients of order $k$ are all equal and satisfying $\forall i \in \llbracket -1, k-1 \rrbracket, L^{(i)} = A^{(i)}$.