Constructing the Field of Values of Decomposable and General Matrices
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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 HK(t) = cos(t) H + sin(t) K. Here HK(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 HK(t), the algorithm decomposes the given dense matrix A conformally into diagonal blocks as HK in U^*AU = 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.
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