Cholesky decomposition is a crucial tool in numerical analysis and scientific computation, offering an efficient method to decompose symmetric and positive definite matrices. This technique is widely used in solving linear equations, least squares problems, and probabilistic modeling, among other applications. However, many resources tend to either lack in-depth explanations of Cholesky decomposition or focus solely on its applications without a clear understanding of the foundational concepts.

This book is designed to fill that gap. It systematically covers everything from the mathematical foundations to algorithms, implementation, and applications, with the goal of helping readers develop a deeper understanding of Cholesky decomposition and apply it to real-world problems. In particular, this book addresses various numerical issues and optimization techniques that may arise during practical computation, aiming to minimize the difficulties readers may encounter when performing actual calculations.

While Cholesky decomposition is a powerful technique, it also has limitations, functioning properly only under certain conditions. The latter part of this book discusses these limitations, ways to address them, and various methods to extend or modify Cholesky decomposition to solve more complex problems.

This book is primarily intended for readers with a basic understanding of Cholesky decomposition, and advanced topics will be covered in subsequent volumes. I hope that through this book, readers will gain a clear understanding of the concept and practical use of Cholesky decomposition and develop the confidence to apply it in real-world scenarios.