A matrix problem over a central quadratic skew field extension

Let F⊂G=F(u) be a central quadratic skew field extension (such that the generator u is central in G) and a natural (G,G)-bimodule. We deal with the matrix problem on finding a canonical form for rectangular matrices over W with help of left elementary transformations of their rows and right elementary transformations of columns over G. We solve this problem reducing it in the separable (resp. inseparable) case to the semilinear (resp. pseudolinear) pencil problem.