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On the Convexity of the Upper Bild
Abstract
We give a proof of the convexity of the upper bild of a quaternionic matrix, a theorem of So and Thompson. After reduction to 2×2 matrices, we introduce real coordinates on a sphere and obtain two representations of the upper bild over the closed unit ball, involving a convex function and a concave function. Minimizing the first and maximizing the second over parameters with a fixed real part yield a convex lower boundary function and a concave upper boundary function, from which convexity follows.
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BibTeX
@misc{carvalho2026convexity,
title = {{On the Convexity of the Upper Bild}},
author = {Luís Carvalho},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18096},
eprint = {2610.18096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18096},
}
RIS
TY - UNPB
TI - On the Convexity of the Upper Bild
AU - Luís Carvalho
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18096
UR - https://arxiv.org/abs/2610.18096
AB - We give a proof of the convexity of the upper bild of a quaternionic matrix, a theorem of So and Thompson. After reduction to 2×2 matrices, we introduce real coordinates on a sphere and obtain two representations of the upper bild over the closed unit ball, involving a convex function and a concave function. Minimizing the first and maximizing the second over parameters with a fixed real part yield a convex lower boundary function and a concave upper boundary function, from which convexity follows.
ER -