27th Conference of the International Linear Algebra Society (ILAS 2026)

Revisiting the Upper Bild Convexity of Quaternionic Numerical Range

May 19, 2026, 2:25 PM

25m

Goodwin Hall 145

Minisymposium Talk Matrix Inequalities, Matrix Equations, and Their Applications Matrix Inequalities, Matrix Equations, and Their Applications

Speaker

Jie Tian (University of Nevada, Reno)

Description

The quaternionic numerical range of a matrix is generally nonconvex, in contrast to the classical complex case. Nevertheless, a theorem of So and Thompson in 1996 asserts that the associated \emph{upper bild} in the complex upper half-plane is always convex.

The original proof of So and Thompson relies on a detailed case-by case and computationally involved analysis, including a reduction to the $2\times 2$ case and an explicit description of boundary curves. A recent note by Kumar in 2019 proposes a shorter and more conceptual argument.

In this talk, we revisit Kumar's approach and identify several gaps in the argument, showing that certain steps require additional justification. These issues point to the importance of incorporating geometric structure into the analysis of the upper bild. We also discuss ongoing work exploring algebraic and geometric features of the upper bild toward a more conceptual understanding of its convexity.

[1] W. So and R. C. Thompson, Convexity of the upper complex plane part of the numerical range of a quaternionic matrix, Linear Multilinear Algebra 41 (1996) 303--362.

[2] P. S. Kumar, A note on convexity of sections of quaternionic numerical range, Linear Algebra Appl. 572 (2019) 92--116.