# Matrix Apportionment: General Cases and Apportionment Constants

## TL;DR

This paper investigates the conditions under which matrices can be transformed into a form with equal modulus entries, classifies certain small matrices, and explores the set of apportionment constants.

## Contribution

It establishes new results on apportionability for nilpotent and low-rank matrices, classifies matrices of order 2 and some of order 3, and initiates the study of apportionment constants.

## Key findings

- All nilpotent matrices are apportionable.
- Matrices with rank at most half their order are apportionable.
- Classification of apportionable matrices of order 2 and some of order 3.

## 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.