Computational Linear Algebra and Tensor Methods for AI: Stability, Randomization, Low-Rank Structure, and Scalable Tensor Computation

Modern AI is built on numerical linear algebra at enormous scale. Matrix products, least-squares problems, eigensolvers, low-rank approximations, tensor contractions, iterative methods, and mixed-precision computation are not peripheral implementation details: they determine whether training is stable, whether inference is efficient, and whether increasingly large models can be represented and computed at all.

Computational Linear Algebra and Tensor Methods for AI develops these ideas from a unified mathematical and computational perspective. The book connects conditioning, backward and forward error, numerical stability, Krylov methods, randomized algorithms, low-rank matrix structure, tensor decompositions, tensor networks, and communication-aware computation to the practical architecture of contemporary AI systems. The central question is not only whether an algorithm is mathematically correct, but how its numerical behavior, memory movement, precision, structure, and approximation error propagate into model training and inference.

Designed as a graduate-level text, the volume emphasizes geometric and computational intuition alongside rigorous analysis. Classical results are presented in forms that make their relevance to large language models and modern accelerator-based computing explicit, including attention, optimizer linear algebra, low-rank adaptation and compression, tensorized representations, mixed precision, and distributed computation.

The result is a bridge between numerical mathematics and AI engineering: a framework for understanding when large-scale linear and multilinear computations are reliable, scalable, and structurally exploitable—and where their mathematical limits begin.