# A short note on the order of the Zhang-Liu matrices over arbitrary fields

## TL;DR

This paper characterizes when Zhang-Liu matrices are diagonalizable over any field, provides their eigen-decomposition, and calculates their order, extending previous results to arbitrary fields.

## Contribution

It offers necessary and sufficient conditions for diagonalizability and eigen-decomposition of Zhang-Liu matrices over arbitrary fields, generalizing earlier work.

## Key findings

- Criteria for diagonalizability over arbitrary fields
- Explicit eigen-decomposition formulas
- Calculation of matrix order in general fields

## Abstract

We give necessary and sufficient conditions for the Zhang-Liu matrices to be diagonalizable over arbitrary fields and provide the eigen-decomposition when it is possible. We use this result to calculate the order of these matrices over any arbitrary field. This generalizes a result of the second author.