Involutory matrices, modulo m
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Abstract
Given the prime power factorization of a positive integer m, a method for calculating the number of all distinct n x n - involutory matrices (mod m) is derived. This is done by first developing a method for the construction and enumeration of involutory matrices (mod Pα), without duplication, for each prime power modulus Pα. Using these results, formulas for the number of distinct involutory matrices (mod Pα) of order n are given where p is an odd prime, p=2, α= 1 and α > 1.
The concept of a fixed group associated with an involutory matrix (mod Pα) is used to characterize such matrices. Involutory matrices (mod Pα) of order n are considered as linear transformations on a vector space of n-tuples to provide uncomplicated proofs for the basic results concerning involutory matrices over a finite field.