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Applications of the joint angular field of values


Author: George Phillip Barker
Journal: Proc. Amer. Math. Soc. 91 (1984), 331-335
MSC: Primary 15A60
DOI: https://doi.org/10.1090/S0002-9939-1984-0744623-X
MathSciNet review: 744623
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Abstract: 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$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0744623-X
Keywords: Numerical range, positive semidefinite matrices, linear transformations of hermitian matrices
Article copyright: © Copyright 1984 American Mathematical Society

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