Moment of a subspace and joint numerical range

For a subspace S of and a fixed basis, we study the compact and convex set that we call the moment of S, where . This set is relevant in the determination of minimal hermitian matrices ( such that for every diagonal D and the spectral norm ). We describe extremal points and certain curves of in term...

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Bibliographic Details
Published in:Linear & multilinear algebra 2023-06, Vol.71 (9), p.1470-1503
Main Authors: Klobouk, Abel H., Varela, Alejandro
Format: Article
Language:English
Subjects:
best approximation
convexity
diagonal matrix
extremal points
joint numerical range
minimal hermitian matrix
Moment of a subspace
Subspaces
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Summary:For a subspace S of and a fixed basis, we study the compact and convex set that we call the moment of S, where . This set is relevant in the determination of minimal hermitian matrices ( such that for every diagonal D and the spectral norm ). We describe extremal points and certain curves of in terms of principal vectors that minimize the angle between S and the coordinate axes of the fixed basis. We also relate to the joint numerical range W of n rank one hermitian matrices constructed with orthogonal projection and the fixed basis used. This connection provides a new approach to the description of and to minimal matrices. As a consequence, the intersection of two of these joint numerical ranges corresponding to orthogonal subspaces allows the construction or detection of a minimal matrix.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2022.2064967