# An algebra for the propagation of errors in matrix calculus

## TL;DR

This paper develops an algebraic framework for error propagation in matrix calculus using external numbers, extending classical properties and addressing exceptions with counterexamples and adaptations.

## Contribution

It introduces an algebra of external numbers for matrix calculus, providing new rules and insights for error analysis in matrices.

## Key findings

- Classical matrix properties mostly hold with modifications
- Counterexamples highlight limitations of traditional rules
- New concepts for near inverses and generalized rank are proposed

## Abstract

We assume that every element of a matrix has a small, individual error, and model it by an external number, which is the sum of a nonstandard real number and a neutrix, the latter being a convex (external) set having the group property. The algebra of external numbers formalizes common error analysis, with rules for calculation which are a sort of mellowed form of the axioms for real numbers. We extend the algebra of external numbers to matrix calculus. Many classical properties continue to hold, sometimes stated in terms of inclusion instead of equality. There are notable exceptions, for which we give counterexamples and investigate suitable adaptations. In particular we study addition and multiplication of matrices, determinants, near inverses, and generalized notions of linear independence and rank.