On the geometry of the envelope of a matrix

The envelope, E(A)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)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)E(A) contains the 2-rank numerical range of A.