Group-Pattern Matrices

This mathematical monograph uses the concept of a group-pattern throughout to conveniently characterize various matrices of unusual historical interest. While a reader should be able to add and multiply matrices in a suitable context, a prior knowledge about groups is not needed. Numerous multiplication tables for sets of n objects are presented in Chapter 1. When the formal definition of a group appears in Chapter 2, it has been well motivated. A group-pattern for a group G that has n elements is provided by the n x n interior of any multiplication table for G in which the identity element of G occupies each of the n principal diagonal positions. An n x n group-pattern matrix results when each group element in the group-pattern is replaced by some element of a given set. If R is the set of all possible n x n group-pattern matrices for G with L that have complex numbers as components, then R has the properties of a ring with respect to matrix addition and multiplication. In addition to the n x n zero matrix, R also contains the n x n identity matrix for multiplication. Let A be an n x n group-pattern matrix for G and L that has variables as components with n distinct variables in its first row. A computer algebra program is included such that: when the input is A and the computer has sufficient capacity, the output specifies all of the automorphisms for G. It runs efficiently on today's typical personal computers when n does not exceed 11. The determinant of the preceding matrix A is expressible as a product of irreducible polynomial combinations of the variables with complex numbers as coefficients. The number of times each irreducible factor appears is equal to the total degree of the factor in the variables. Those factors are explicitly given by a computer algebra program when various details about G are known.

• | Author: Roger Chalkley
• | Publisher: Roger Chalkley
• | Publication Date: May 02, 2022
• | Number of Pages: 262 pages
• | Language: English
• | Binding: Hardcover/Computers
• | ISBN-10: 0578287889
• | ISBN-13: 9780578287881