DIFFERENTIAL TOPOLOGY OF NUMERICAL RANGE

The numerical range of an n x n matrix, also known as its field of val ues, is reformulated as the image of a smooth quadratic mapping from t he n - 1 dimensional complex projective: space to the complex plane. T his paper investigates the numerical range from the perspective of dif ferential topology (Morse theory). More specifically, the boundary of the range is interpreted as a rank 1 critical value curve and its shar p points are interpreted as rank 0 critical values. More importantly, I-he map is shown to have additional critical value curves in the inte rior of the numerical range. These additional curves are shown to have such singularity phenomena as cusps and swallow tails, to be the caus tic envelopes of families of lines, and to exhibit the so-called ''nor mal bifurcation'' when an eigenvalue: becomes unitarily decoupled. (C) 1998 Elsevier Science Inc. All rights reserved.