# Filmor Theorem for integers

## TL;DR

This paper presents a simple, explicit two-step algorithm to extend Fillmore's Theorem to integer matrices, allowing the construction of similar matrices with prescribed integer diagonals.

## Contribution

It introduces a straightforward, two-step similarity algorithm that extends Fillmore's Theorem to integer matrices, simplifying previous methods.

## Key findings

- The algorithm completes in only two similarity steps.
- It guarantees the existence of an integer similar matrix with any prescribed diagonal summing to the trace.
- The method simplifies the construction process for integer matrices.

## Abstract

Fillmore Theorem says that if $A$ is a nonscalar matrix of order $n$ over a field $\mathbb{F}$ and $\gamma_1,\ldots,\gamma_n\in \mathbb{F}$ are such that $\gamma_1+\cdots+\gamma_n=\text{tr} \, A$, then there is a matrix $B$ similar to $A$ with diagonal $(\gamma_1,\ldots,\gamma_n)$. Fillmore proof works by induction on the size of $A$ and implicitly provides an algorithm to construct $B$. We develop an explicit and extremely simple algorithm that finish in two steps (two similarities), and with its help we extend Fillmore Theorem to integers (if $A$ is integer then we can require to $B$ to be integer).