Differential Topology of Numerical Range

Abstract: The numerical range of an n x n matrix, also known as its field of values, is reformulated as the image of a smooth quadratic mapping from the n-1 dimensional complex projective space to the complex plane. We investigate the numerical range from the perspective of differential topology (Morse theory). More specifically, the boundary of the range is interpreted as a rank 1 critical value curve and its sharp points are interpreted as rank 0 critical values. More importantly, the map is shown to have additional critical value curves in the interior of the numerical range. These additional curves are shown to have such singularity phenomena as cusps and swallow tails, to be the caustic envelopes of families of lines, and to exhibit the so-called ``normal bifurcation'' when an eigenvalue becomes unitarily decoupled.

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