Let $A \in \mathcal{M}_2(\mathbb{R})$. a) Determine the matrices whose eigenvalue circle has zero radius and characterize geometrically their canonically associated endomorphism. b) When the eigenvalue circle is reduced to its center, specify the canonically associated endomorphism, on the one hand when this center belongs to the unit circle (with center the origin $O=(0,0)$ and radius 1) and on the other hand when this center belongs to the $x$-axis. c) What can be said about the matrix $A$ and $f_A$ when the eigenvalue circle $\mathcal{CP}_A$ has zero radius and center belonging to the $y$-axis $\{0\} \times \mathbb{R}$?
Let $A \in \mathcal{M}_2(\mathbb{R})$.
a) Determine the matrices whose eigenvalue circle has zero radius and characterize geometrically their canonically associated endomorphism.
b) When the eigenvalue circle is reduced to its center, specify the canonically associated endomorphism, on the one hand when this center belongs to the unit circle (with center the origin $O=(0,0)$ and radius 1) and on the other hand when this center belongs to the $x$-axis.
c) What can be said about the matrix $A$ and $f_A$ when the eigenvalue circle $\mathcal{CP}_A$ has zero radius and center belonging to the $y$-axis $\{0\} \times \mathbb{R}$?