Coxeter Groups: Spectral Theorem and Sylvester's Criterion

This lecture covers the Sylvester criterion for positive definite matrices and a recursive formula for the determinant of a marked graph. It explores the classification of connected positive definite Coxeter graphs, the spectral theorem for real symmetric matrices, and the properties of eigenvalues. The lecture also delves into the concept of Coxeter graphs, their principal minors, and subgraphs. Additionally, it discusses the Lemma related to the number of possible eigenvalues of a matrix based on the positivity of inner products. The lecture concludes with examples of positive definite Coxeter graphs and their determinants, emphasizing the application of the spectral theorem and Sylvester's criterion.