Date of Award
Summer 2026
Document Type
Thesis
Degree Name
Master of Science (MS) in Mathematics
Department
Mathematical, Computing & Information Sciences
Committee Chair
Jaedeok Kim
Abstract
The Elliptical Range Theorem states that the numerical range of a 2x2 complex matrix A is an elliptical disc with foci at the eigenvalues of A and center at tr(A)/2. Many proofs have been offered for this theorem. In this paper, we provide another one. The strategy we will use is to decompose the matrix A into the sum of rank-1 operators of the form YxX*, where X and Y are in 2-D complex vectors. When analyzing the numerical range of a rank-1 operator of this form, three results can happen: i) the numerical range forms a disc at the origin, ii) the numerical range forms a line from the origin to , or iii) the numerical range forms an ellipse with foci at the origin and at . The numerical range of A, therefore, assumes one of those shapes up to translation and degeneration.