The Cartan–Dieudonné–Scherk theorems for complex S-orthogonal matrices

Let Mn(C)Mn(C) be the set of all n-by-n   matrices with complex entries. Let Sn+ be the set of all nonsingular symmetric matrices in Mn(C)Mn(C), let Sn− be the set of all nonsingular skew-symmetric matrices in Mn(C)Mn(C), and let Sn=Sn+∪Sn−. Let S∈SnS∈Sn be given. An A∈Mn(C)A∈Mn(C) is called S-orthogonal   if ATSA=SATSA=S. Let OSOS be the set of all S  -orthogonal matrices in Mn(C)Mn(C). An H∈OSH∈OS is called a symmetry   if rank(H−I)=1rank(H−I)=1. Let HSHS be the set of all symmetries in OSOS. We show that every Q∈OSQ∈OS is a product of elements of HSHS. If Q=IQ=I, then Q   is a product of two elements of HSHS. Suppose that rank(Q−I)=m≥1rank(Q−I)=m≥1. If S(Q−I)S(Q−I) is not skew-symmetric, then Q can be written as a product of m   elements of HSHS and Q cannot be written as a product of less than m   elements of HSHS. If S(Q−I)S(Q−I) is skew-symmetric and if S∈Sn+, then Q   can be written as a product of m+2m+2 elements of HSHS and Q   cannot be written as a product of less than m+2m+2 elements of HSHS. If S(Q−I)S(Q−I) is skew-symmetric and if S∈Sn−, then Q   can be written as a product of m+1m+1 elements of HSHS and Q   cannot be written as a product of less than m+1m+1 elements of HSHS.