Analytical and Computational Study of Numerical Domain of Matrices

Abstract

This scientific article presents the results of research on Numerical Domain of Complex Matrices. In the first part, we address the main properties and their respective demonstrations, highlighting the P8 property on convexity of the numerical domain, demonstrated through the Toeplitz-Hausdorff Theorem. In the second part, we show how to determine the numerical domain of a matrix, in addition to making its graphic representation in the complex plane, for this purpose, we use the elliptical domain theorem, because when represented in the complex plane, the numerical domain reveals valuable information about the behavior of the matrix eigenvalues ​​and their associated geometry. The third and final part of the work is to determine the numerical domain of an arbitrary matrix, with its respective eigenvalues, through numerical approximations. To this end, we implemented, through the MATLAB program, an algorithm proposed by Charles R. Johnson.