On the Formalization of Gram-Schmidt Process for Orthonormalizing a Set of Vectors

Abstract

In this article, we formalize the Gram-Schmidt process in the Mizar system [2], [3] (compare another formalization using Isabelle/HOL proof assistant [1]). This process is one of the most famous methods for orthonormalizing a set of vectors. The method is named after Jørgen Pedersen Gram and Erhard Schmidt [4]. There are many applications of the Gram-Schmidt process in the field of computer science, e.g., error correcting codes or cryptology [8]. First, we prove some preliminary theorems about real unitary space. Next, we formalize the definition of the Gram-Schmidt process that finds orthonormal basis. We followed [5] in the formalization, continuing work developed in [7], [6].