A Coxeter matrix is a square matrix associated with a Coxeter group, where each entry represents the order of the product of two generators of the group. It is symmetric, with diagonal elements equal to one, and off-diagonal elements that are either integers greater than one or infinity, indicating the absence of a relation between the corresponding generators.

Coxeter Matrix

A square matrix is a matrix with the same number of rows and columns, often used to represent linear transformations in vector spaces. Its properties, such as determinant, eigenvalues, and trace, are foundational in linear algebra and have significant applications in various scientific fields.

Square Matrix

A Coxeter group is an abstract group defined by a set of generators subject to specific relations, primarily involving products of pairs of generators that are involutions. These groups are fundamental in the study of symmetries and have applications in various fields, including geometry, algebra, and combinatorics.

Coxeter Group

Order of the product

Two generators

A symmetric matrix is a square matrix that is equal to its transpose, meaning the element at row i, column j is the same as the element at row j, column i. This property makes symmetric matrices particularly important in linear algebra, as they often have real eigenvalues and orthogonal eigenvectors, simplifying many mathematical computations.

Symmetric Matrix

Diagonal elements in a matrix are the entries that extend from the top left to the bottom right corner, forming a straight line. These elements are crucial in determining properties like the trace, determinant, and eigenvalues of a matrix, and play a significant role in linear algebra and related computations.

Diagonal Elements

Off-diagonal elements

Integers greater than one

Infinity is a concept in mathematics and philosophy that describes something without any limit or end. It is not a number but rather an idea that helps to understand and describe the unbounded nature of certain mathematical and abstract constructs.

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