Applications of the Joint Angular Field of Values
George Phillip Barker · Proceedings of the American Mathematical Society · 1984
Let ${A_1}, \ldots ,{A_m}$ be $n \times n$ hermitian matrices and let ${\mathcal {H}_n}$ be the real space of $n \times n$ hermitian matrices. If ${\operatorname {span}}\left \{ {{A_1}, \ldots ,{A_m}} \right \} = {\mathcal {H}_n}$, then the extreme rays of the joint angular field of values of $\left \{ {{A_1}, \ldots ,{A_m}} \right \}$ are determined. Then this cone is used to give necessary and sufficient conditions for the existence of hermitian matrices ${B_1}, \ldots ,{B_m}$ such that ${A_1} \otimes {B_1} + \cdots + {A_m} \otimes {B_m}$ preserves the cone of the positive semidefinite matrices where $A \otimes B$ is the dyad product $A \otimes B\left ( H \right ) = \left ( {{\text {tr}}BH} \right )A$.