Matrix Apportionment: General Cases and Apportionment Constants
arXiv:2508.21297
Abstract
A matrix is apportionable if it is similar to a matrix whose entries have equal moduli. This paper shows that all nilpotent matrices and all matrices with rank at most half their order are apportionable. General results are established and applied to classify all apportionable matrices of order 2 and partially those of order 3. Additionally, the study of the set of apportionment constants for matrices is initiated.