A new polar decomposition in a scalar product space

There are various definitions of right and left polar decompositions of an m×n matrix F∈Km×n (where K=C or R) with respect to bilinear or sesquilinear products defined by nonsingular matrices M∈Km×m and N∈Kn×n. The existence and uniqueness of such decompositions under various assumptions on F, M, and N have been studied. Here we introduce a new form of right and left polar decompositions, F=WS and F=S′W′, respectively, where the matrix W has orthonormal columns (W′ has orthonormal rows) with respect to suitably defined scalar products which are functions of M, N, and F, and the matrix S is selfadjoint with respect to the same suitably defined scalar products and has eigenvalues only in the open right half-plane. We show that our right and left decompositions exist and are unique for any nonsingular matrices M and N when the matrix F satisfies (F[M,N])[N,M]=F and F[M,N]F (FF[M,N], respectively) is nonsingular, where F[M,N]=N−1F#M with F#=FT for real or complex bilinear forms and F#=F¯T for sesquilinear forms. When M=N, our results apply to nonsingular square matrices F. Our assumptions on F, M, and N are in some respects weaker and in some respects stronger than those of previous work on polar decompositions.