Involutory matrices over finite commutative rings

Abstract

An n × n matrix A is called involutory iff A 2= I n , where I n is the n × n identity matrix. This paper is concerned with involutory matrices over an arbitrary finite commutative ring R with identity and with the similarity relation among such matrices. In particular the authors seek a canonical set C with respect to similarity for the n × n involutory matrices over R—i.e., a set C of n × n involutory matrices over R with the property that each n × n involutory matrix over R is similar to exactly on matrix in C . Because of the structure of finite commutative rings and because of previous research, they are able to restrict their attention to finite local rings of characteristic a power of 2, and although their main result does not completely specify a canonical set C for such a ring, it does solve the problem for a special class of rings and shows that a solution to the general case necessarily contains a solution to the classically unsolved problem of simultaneously bringing a sequence A 1,…, A v of (not necessarily involutory) matrices over a finite field of characteristic 2 to canonical form (using the same similarity transformation on each A i ). (More generally, the authors observe that a theory of similarity fot matrices over an arbitrary local ring, such as the well-known rational canonical theory for matrices over a field, necessarily implies a solution to the simultaneous canonical form problem for matrices over a field.) In a final section they apply their results to find a canonical set for the involutory matrices over the ring of integers modulo 2 m and using this canonical set they are able to obtain a formula for the number of n × n involutory matrices over this ring.