Advanced linear algebra (2506.18666v1)

Abstract: This is an introduction to advanced linear algebra, with emphasis on geometric aspects, and with some applications included too. We first review basic linear algebra, notably with the spectral theorem in its general form, and with the theory of the resultant and discriminant. Then we discuss the Jordan form and its basic applications to physics, and other advanced decomposition results for the matrices. We then go into positivity topics, involving matrices and bilinear forms, and with a look into curved space-time, and discrete Laplacians. Finally, we discuss the various groups of matrices, with a look at reflection groups, Lie groups, spin matrices, and random matrices.

• The paper presents a comprehensive synthesis of algebraic and geometric techniques, emphasizing core topics like vectors, matrix theory, and spectral properties.
• It details advanced matrix decompositions such as Jordan form and singular value decomposition, highlighting their use in modeling physical dynamics.
• The work connects bilinear forms, matrix groups, and multivariable calculus to practical applications in physics and geometry.

Advanced Linear Algebra: A Comprehensive Overview

This book provides an introduction to advanced linear algebra, emphasizing its geometric facets and practical applications. It balances algebraic rigor with geometric intuition, aiming to provide readers with a dynamic understanding of linear algebra concepts.

Core Structure and Organization

The book is structured into four parts, each comprising four chapters, with each chapter spanning 24 pages. Each chapter is further divided into four sections, followed by an exercise section. Including front and back matter, the book totals 400 pages. The four parts are organized with an emphasis on algebra and symmetry.

Part I: Foundations and Basic Theory

This section revisits fundamental linear algebra concepts, including vectors, linear maps, matrices, matrix inversion, determinants, eigenvectors, eigenvalues, and diagonalization. It also covers advanced topics such as the spectral theorem and the theories of the resultant and discriminant. The density of diagonalizable matrices over $\mathbb C$ is also noted.

Part II: Advanced Matrix Decompositions

This part addresses matrices that are not diagonalizable, exploring the Jordan form and its applications in physics, particularly in dynamical systems. Advanced decomposition methods such as singular value decomposition are discussed, including their implications for infinite-dimensional spaces and compact operators.

Part III: Positivity and Bilinear Forms

This section focuses on the concepts of positivity and negativity, driven by the eigenvalues of matrices, especially Hessian matrices. The classification of bilinear forms is discussed, and connects these mathematical structures to curved spacetime and Lorentz geometry. Topics include bistochastic matrices, discrete Fourier analysis, designs, and discrete Laplacians.

Part IV: Matrix Groups and Applications

The final section is dedicated to matrix groups. It explores reflection groups and their relation to army formations, Lie groups and their role as the Navy, physics groups, and random matrices as the Air Force.

Geometric and Physical Underpinnings

The author emphasizes a dynamic perspective on linear algebra, viewing vectors and linear maps as objects with underlying physical significance. The book consistently underscores the importance of geometry and physics in understanding and applying linear algebra concepts.

Complex Numbers and Fields

The book explores linear algebra over the complex numbers, highlighting the diagonalization of rotation matrices and the use of roots of unity. It also explores linear algebra over arbitrary fields, including finite fields and p-adic numbers, discussing their properties and implications for geometric constructions.

Multivariable Calculus

The book discusses the relationship between matrices and multivariable calculus. It emphasizes the use of matrices in derivatives, partial derivatives, Taylor formulas, and Hessian matrices.

Concluding Remarks

The book synthesizes algebraic techniques with geometric insights, offering a comprehensive view of advanced linear algebra. By covering a range of topics, including matrix decompositions, bilinear forms, and matrix groups, the book provides a solid foundation for further exploration in various mathematical and scientific domains.