We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$. We seek to determine the set of matrices $R \in \mathcal{M}_3(\mathbb{C})$ such that $R^2 = A$. We denote by $\rho$ the endomorphism canonically associated with $R$.
Deduce the set of square roots of $A$. One may consider $R' = P^{-1}RP$.