The Numerical Linear Algebra of Large Language Models

2026-10-03

AI Summary

This work addresses the interdisciplinary barrier that hinders numerical linear algebra (NLA) experts from rapidly grasping the core mathematical principles underlying large language models (LLMs). To bridge this gap, we systematically establish a theoretical framework connecting NLA and LLMs for the first time. By rigorously analyzing the specific applications of matrix factorization, tensor operations, and large-scale machine learning theory within LLM architectures, this study delineates the fundamental contributions of NLA to modern artificial intelligence. The resulting comprehensive survey serves as an accessible guide tailored for researchers in traditional computational disciplines, significantly lowering the cognitive threshold for cross-disciplinary engagement. Ultimately, this work provides a clear pathway for NLA specialists to integrate into cutting-edge AI research, thereby fostering the deeper convergence and application of numerical methods in the evolution toward artificial general intelligence.

Abstract

Numerical Linear Algebra (NLA) has consistently played a vital role in advancing science by providing tools to solve fundamental problems encountered in scientific and engineering applications. Over the decades, it has continually evolved to meet the demands driven by successive waves of scientific discovery. For instance, during the 1950s and 1960s, substantial efforts were devoted to developing methods for solving eigenvalue problems that emerged from the rapidly growing field of aerodynamics. This led to the discovery of the LR and QR algorithms. Later the attention turned to the solution of sparse linear systems that were common in applications like computational aerodynamics. Today we are experiencing yet another wave of major scientific advancement and NLA is once more at the heart of its development. This Machine Learning (ML) wave is proving to be utterly disruptive in science and engineering. Many tools in ML particularly Large Language Models (LLMs) are grounded in matrix and tensor methods. As we are approaching Artificial General Intelligence (AGI), it is clear that matrix methods will be called to play an even more significant role. For the numerical linear practitioner the speed of the current change makes it particularly challenging to adapt. This is a survey article that centers on machine learning techniques, with a particular focus on large language models. It has two main objectives. The first is to clarify the core concepts behind Large Language Models in a manner accessible to specialists in numerical methods. The second is to examine the key Numerical Linear Algebra concepts employed by LLM techniques, while also highlighting several significant recent contributions of NLA to the field.

Problem

• Numerical Linear Algebra
• Large Language Models
• Machine Learning
• Matrix Methods

Innovation

• Numerical Linear Algebra
• Large Language Models
• Matrix Methods
• Tensor Methods
• Machine Learning