Quaternionic numerical range of complex matrices
Lu\'is Carvalho, Cristina Diogo, S\'ergio Mendes
TL;DR
This paper investigates the quaternionic numerical range of complex matrices, establishing conditions for its convexity and its relation to the complex numerical range, along with a proof of a modified conjecture.
Contribution
It introduces a new characterization of the quaternionic numerical range based on the complex numerical range and real values, and proves a modified conjecture by So and Tompson.
Findings
The shape of the quaternionic numerical range depends on the complex numerical range and two real values.
Conditions are identified when the bild of a complex matrix matches its complex numerical range.
The paper determines when the quaternionic numerical range is convex.
Abstract
The paper explores further the computation of the quaternionic numerical range of a complex matrix. We prove a modified version of a conjecture by So and Tompson. Specifically, we show that the shape of the quaternionic numerical range for a complex matrix depends on the complex numerical range and two real values. We establish under which conditions the bild of a complex matrix coincides with its complex numerical range and when the quaternionic numerical range is convex.
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