Question 132
Consider $A, B \in M_n(\mathbb{R})$, and the function $f: M_n(\mathbb{C}) \to M_n(\mathbb{C})$, defined by $f(Z) = AZ + B\bar{Z}$, $Z \in M_n(\mathbb{C})$, where $\bar{Z}$ is the matrix having as entries the conjugates of the entries of $Z$. Prove that the following are equivalent:
(1) $f$ is injective;
(2) $f$ is surjective;
(3) matrices $A+B$ and $A-B$ are non-singular.
(1) $f$ is injective;
(2) $f$ is surjective;
(3) matrices $A+B$ and $A-B$ are non-singular.