Constructing the Field of Values of Decomposable and General Matrices.
Frank Uhlig · arXiv (Cornell University) · 2020
This paper describes and develops a fast and accurate algorithm that computes the field of values boundary curve for every conceivable complex square matrix $A$, i.e., for matrices with or without repeated eigenvalues and with or without Jordan block structures. It relies on a matrix flow decomposition method that finds the coarsest block diagonal flow representation for the hermitean matrix flow $H\!K(t) = \cos(t) H + \sin(t) K$. Here $H\!K(t)$ is a 1-parameter varying linear combination of the real and skew part matrices $H = (A+A^*)/2$ and $K = (A-A^*)/(2i)$ of $A$. For decomposing flows $H\!K(t)$, the algorithm decomposes the given dense matrix $A$ conformally into diagonal blocks as $H\!K$ in $U^*AU = \text { diag} (A_j)$ for a unitary matrix $U$. It then computes the field of values boundaries separately for each diagonal block $A_j$ using the fast ZNN parameter varying flow eigenvalue method. Finally it saves the convex hull of all intermediate field of values boundary curves in order to plot the field of values of $A$ properly. The algorithm removes standard restrictions for path following FoV methods that generally cannot deal with decomposing matrices $A$ due to possible eigencurve crossings for such $A$. Tests and numerical comparisons are included.