On the geometry of the envelope of a matrix

The envelope, E(A), of a complex square matrix A is a region in the complex plane that contains the spectrum of A and is contained in the numerical range of A. The envelope is compact but not necessarily convex or connected. The connected components of E(A) have the potential of isolating the eigenvalues of A, leading us to study its geometry, boundary, and number of components. We also examine the envelope of normal matrices and similarities. In the process, we observe that E(A) contains the 2-rank numerical range of A