The eigenblock equation for a Fockian-or Huckel-type Hamiltonian matrix of a molecule being an alternative form of the Brillouin theorem and determining Ε½ . the noncanonical molecular orbitals NCMOs was generalized to the case of nonorthogonal initial basis sets. As in the Lowdin's partitioning technique, the initial set ΓΆf basis orbitals was divided into two subsets. This allowed the problem to be formulated in terms of these subsets and the respective submatrices. A perturbative approach was developed for the solution of the new matrix equation in terms of entire submatrices Ε½ . blocks of both Fockian and overlap matrices without specifying either the structures or Ε½ . dimensions of these submatrices. Two multidimensional molecular orbitals MMOs Ε½ . containing the subsets of occupied and vacant noncanonical MOs NCMOs , respectively, were defined for any molecule, and these MMOs were expressed in the form of linear combinations of two subsets of basis orbitals with matrix coefficients. A linear combination of this type may be considered as a generalization of the usual LCAO approximation for an MO of a two-level system. The relevant common form of the bond-order matrix was shown to be obtainable on the basis of the projector to a single occupied MMO. These results allowed us to conclude that the generalized matrix problem obtained in this article forms the basis of the noncanonical MO theory of electronic structures of molecules in its matrix representation. The new matrix problem along with its solution was compared to related problems and methods.