The Numerical Linear Algebra of Large Language Models
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.
Related
- Advanced Linear Algebra with Applications - Part I (Numerical linear algebra for PDEs, machine learning, and data assimilation)
- Randomized algorithms in numerical linear algebra for scientific computing
- Inverse problems with fractional operators: an analytical study and its numerical linear algebra treatment via the GLT theory
- GPU-Enabled Large-Scale Optimization Using Randomized Linear Algebra
- When and Why Do Linear Bias Probes Fail? A Geometric and Statistical Theory of Bias Detectability in Large Language Model Representations
BibTeX
@misc{baggag2026numerical,
title = {{The Numerical Linear Algebra of Large Language Models}},
author = {Abdelkader Baggag and Yousef Saad},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04631},
eprint = {2610.04631},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04631},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Numerical Linear Algebra of Large Language Models
AU - Abdelkader Baggag
AU - Yousef Saad
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04631
UR - https://arxiv.org/abs/2610.04631
AB - 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.
ER -