Lecture

Coxeter Groups: Spectral Theorem and Sylvester's Criterion

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In course

MATH-335: Coxeter groups

Study groups generated by reflections ...

Description

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.

Instructor

Anna Lachowska

Teaching & PhD Courses Algebra MATH-310 This is an introduction to modern algebra: groups, rings and fields. Analysis I MATH-101(e) We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable. Analysis II MATH-106(e) The course studies fundamental concepts of analysis and the calculus of functions of several variables. Representation theory I - finite groups MATH-314 This is a standard c

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Mathematics

Algebra: Linear algebra

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In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for arises in this fashion.

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Orthogonal basis

In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis. Any orthogonal basis can be used to define a system of orthogonal coordinates Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds.

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