Products of central collineations

AbstractAn 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 xi ∈ F such that x1⋯xn=detM, then there exist central matrices Mi such that M=M1⋯Mn and xi=detMi 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)