Products of central collineations

An n by n matrix M over a (commutative) field F is said to be central if M ? I has rank 1. We say that M is an involution if M²=I; if M is also central we call M a simple involution. We will prove that any n-by-n matrix M satisfying detM=±1 is the product of n+2 or fewer simple involutions. This can be reduced to n+1 if F contains no roots of the equation xⁿ=(?1)ⁿ other than ±1. Any ordered field is of this kind. Our main result is that if M is any n-by-n nonsingular nonscalar matrix and if x_i ∈ F such that x₁?x_n=detM, then there exist central matrices M_i such that M=M₁?M_n and x_i=detM_i for i=1,…,n. We will apply this result to the projective group PGL(n,F) and to the little projective group PSL(n,F).