The ϕS polar decomposition of matrices

Let S∈M2n be skew-symmetric and nonsingular. For X∈M2n, we show that the following are equivalent: (a) X has a ϕS polar decomposition, (b) rank([XϕS(X)]i)=rank([ϕS(X)X]i) and rank([XϕS(X)]iX) is even for all nonnegative integers i, and (c) XϕS(X) is similar to ϕS(X)X and rank([XϕS(X)]iX) is even for all nonnegative integer i.