Extended Gram-Schmidt process on sesquilinear spaces over finite fields

Year 2025, Early Access, 1 - 14

Yasin Prasetia Rian Kurnia Andriko Andriko Pudji Astuti

https://doi.org/10.24330/ieja.1718902

Abstract

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.

Keywords

Finite field , sesquilinear product , self-orthogonal , orthonormal subset , Gram-Schmidt process

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