# A Note on Adjoint Linear Algebra

## TL;DR

This paper introduces a novel proof for adjoint linear systems based on Algorithmic Differentiation, extending to higher-order systems and providing a new perspective on adjoint operations in linear algebra.

## Contribution

It presents a new proof method for adjoint linear systems using Algorithmic Differentiation principles, applicable to various matrix operations and higher-order systems.

## Key findings

- New proof for adjoint systems based on Algorithmic Differentiation
- Extension to higher-order adjoint linear systems
- Alternative proof for matrix-matrix and vector products

## Abstract

A new proof for adjoint systems of linear equations is presented. The argument is built on the principles of Algorithmic Differentiation. Application to scalar multiplication sets the base line. Generalization yields adjoint inner vector, matrix-vector, and matrix-matrix products leading to an alternative proof for first- as well as higher-order adjoint linear systems.