Classification of sesquilinear forms with the first argument on a subspace or a factor space

Let V be a vector space over a field or skew field F, and let U be its subspace. We study the canonical form problem for bilinear or sesquilinear formsU×V→F,(V/U)×V→Fand linear mappings U → V, V → U, V/U → V, V → V/U. We solve it over F=C and reduce it over all F to the canonical form problem for ordinary linear mappings W → W and bilinear or sesquilinear forms W×W→F. Moreover, we give an algorithm that realizes this reduction. The algorithm uses only unitary transformations if F=C, which improves its numerical stability. For linear mapping this algorithm can be derived from the algorithm by Nazarova et al. [L.A. Nazarova, A.V. Roiter, V.V. Sergeichuk, V.M. Bondarenko, Application of modules over a dyad for the classification of finite p-groups possessing an abelian subgroup of index p and of pairs of mutually annihilating operators, Zap. Nauchn. Sem., Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 28 (1972) 69-92, translation in J. Soviet Math. 3 (5) (1975) 636-654].