TITLE:
Geometric Distortion of Numerical Ranges under Similarity and Metric Similarity
AUTHORS:
Faith M. Mwangi, Benard M. Nzimbi, Stephen W. Luketero
KEYWORDS:
Numerical Range, Numerical Radius, Similarity Transformation, Metric Similarity, Condition Number, Geometric Similarity, Similitude, Operator Theory
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.8,
August
21,
2026
ABSTRACT: We study the geometric distortion of numerical ranges of complex matrices under similarity and metric similarity transformations, measuring that distortion through the condition number of the transforming matrix. Four results are obtained. First, we prove a two-sided bound on the numerical radius of a similarity transform, expressed in terms of the condition number, and we give an explicit example showing that the corresponding set inclusion of the numerical range into a dilate of the convex hull of the prototype range fails in general: similarity changes the shape of the numerical range, and not merely its size. Second, for two-by-two nilpotent matrices under diagonal similarity, we obtain an exact scale factor, namely the modulus of the ratio of the two diagonal entries, which shows that the conjectured optimal constant in the first result, the condition number itself, is attained. Third, we prove that the unitary factor in a metric similarity leaves the numerical range unchanged, so that metric similarity and ordinary similarity produce identical numerical-range distortion. Fourth, we resolve completely, in the two-by-two case, the question of when the normalized numerical ranges of the two matrices are related by a rotation: this happens precisely when the matrix is nilpotent, or the two numerical ranges coincide, and the rotation angle can then always be taken to be zero. Throughout, the spectrum is preserved while the numerical range is distorted in a manner controlled, but in general not exactly described, by the condition number.