Extended Gram-Schmidt process on sesquilinear spaces over finite fields

Orthonormal bases play an important role in the geometric study of vector spaces. For inner product spaces over real or complex number fields, we can apply Gram-Schmidt algorithm to construct an orthonormal subset from a linearly independent subset. However, on sesquilinear spaces over finite fields, Gram-Schmidt algorithm fails to produce an orthonormal subset because of the presence of non-zero, self-orthogonal vectors. In fact, there is a subspace that does not contain an orthonormal basis. In this paper, we study sesquilinear spaces over finite fields and show that a non-zero subspace has an orthonormal basis if and only if it is non-degenerate. An Extended Gram-Schmidt Process (EG-SP) is then discussed to construct an orthogonal subset from a linearly independent subset having equal generated subspaces. An advantage of the proposed EG-SP is that the obtained orthogonal subset is orthonormal when the generated subspace is non-degenerate. In addition, we can also extend an orthonormal subset of a sesquilinear space to an orthonormal basis.