Advanced linear algebra

TL;DR

This paper provides a comprehensive introduction to advanced linear algebra, emphasizing geometric perspectives, matrix decompositions, positivity, and group theory, with applications to physics and curved space-time.

Contribution

It offers an integrated overview of advanced linear algebra topics, including spectral theory, Jordan form, positivity, and matrix groups, with applications to physics and geometry.

Findings

Detailed review of spectral theorem and matrix decompositions

Discussion of positivity and applications to curved space-time

Analysis of matrix groups like Lie groups and reflection groups

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.