# Triangular Decomposition of Matrices in a Domain

## TL;DR

This paper introduces deterministic recursive algorithms for matrix triangular decompositions over commutative domains, generalizing LU and Bruhat decompositions with complexity comparable to matrix multiplication.

## Contribution

It presents a unified recursive approach for triangular decompositions in commutative domains, extending classical decompositions like LU and Bruhat.

## Key findings

- Algorithms have the same complexity as matrix multiplication
- Decomposition generalizes LU and Bruhat decompositions
- Applicable to matrices over commutative domains

## Abstract

Deterministic recursive algorithms for the computation of matrix triangular decompositions with permutations like LU and Bruhat decomposition are presented for the case of commutative domains. This decomposition can be considered as a generalization of LU and Bruhat decompositions, because they both may be easily obtained from this triangular decomposition. Algorithms have the same complexity as the algorithm of matrix multiplication.