# Strongly Independent Matrices and Rigidity of $\times A$-Invariant Measures on $n$-Torus

## TL;DR

This paper introduces strongly independent matrices over fields, explores their existence, and applies these matrices over rationals to establish measure rigidity results for endomorphisms on the n-torus.

## Contribution

It defines strongly independent matrices, proves their existence or non-existence over various fields, and applies these findings to measure rigidity on the n-torus.

## Key findings

- Existence of strongly independent matrices over certain fields
- Non-existence over algebraically closed fields
- Measure rigidity results for endomorphisms on the n-torus

## Abstract

We introduce the concept of strongly independent matrices over any field, and prove the existence of such matrices for certain fields and the non-existence for algebraically closed fields. Then we apply strongly independent matrices over rational numbers to obtain a measure rigidity result for endomorphisms on $n$-torus.