An envelope for the spectrum of a matrix

Abstract

We introduce and study an envelope-type region E(A) in the complex plane that contains the eigenvalues of a given n × n complex matrix A. E(A) is the intersection of an infinite number of regions defined by cubic curves. The notion and method of construction of E(A) extend the notion of the numerical range of A, F (A), which is known to be an intersection of an infinite number of half-planes; as a consequence, E(A) is contained in F (A) and represents an improvement in localizing the spectrum of A.

Shell extremal eigenvalues of tridiagonal Toeplitz Matrices

“…e −iθ Γ in (e iθ A). For more information on the envelope, see [3], [4], [14], [15] and the references therein.…”

Ritz Value Localization for Non-Hermitian Matrices

“…(Recently Psarrakos and Tsatsomeros have used the second largest eigenvalue of H to develop inclusion regions for the spectrum [16].) 2.2.…”