Segre Characteristic Equivalence

TL;DR

This paper investigates the number of matrices with the same Segre characteristic by analyzing the possible Jordan Normal Forms for a given matrix dimension, providing insights into their classification.

Contribution

It offers a novel enumeration of Jordan Normal Forms corresponding to a fixed matrix dimension, linking Segre characteristics to matrix similarity classes.

Findings

Derived formulas for counting Jordan Normal Forms

Established relationships between Segre characteristics and matrix classification

Provided bounds on the number of equivalent matrices

Abstract

Given only the dimension, $n$, of a square matrix $A \in M(n,\mathbb{C})$, how many Segre Characteristic equivalent matrices are there? Jordan Normal Form Theorem states that any linear operator over $\mathbb{C}$ is similar to a matrix in Jordan Normal Form. As such, this is a question of counting the number of possible Jordan Normal Forms for a given dimension. So, equivalently, how many Jordan Normal Forms can an $n\times n$ matrix possibly have?