On some 4-by-4 matrices with bi-elliptical numerical ranges
Titas Geryba, Ilya M. Spitkovsky · Linear and Multilinear Algebra · 2020
A complete description of 4-by-4 matrices αICDβI, with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is obtained by reduction to the leading special case in which C−D∗ also is a scalar multiple of the identity. In particular cases when in addition α−β is real or pure imaginary, the results take an especially simple form. An application to reciprocal matrices is provided.