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<p>In this article, we formalize the Gram-Schmidt process in the Mizar system [<xref ref-type="bibr" rid="j_forma-2023-0005_ref_002">2</xref>], [<xref ref-type="bibr" rid="j_forma-2023-0005_ref_003">3</xref>] (compare another formalization using Isabelle/HOL proof assistant [<xref ref-type="bibr" rid="j_forma-2023-0005_ref_001">1</xref>]). 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 [<xref ref-type="bibr" rid="j_forma-2023-0005_ref_004">4</xref>]. There are many applications of the Gram-Schmidt process in the field of computer science, e.g., error correcting codes or cryptology [<xref ref-type="bibr" rid="j_forma-2023-0005_ref_008">8</xref>]. 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 [<xref ref-type="bibr" rid="j_forma-2023-0005_ref_005">5</xref>] in the formalization, continuing work developed in [<xref ref-type="bibr" rid="j_forma-2023-0005_ref_007">7</xref>], [<xref ref-type="bibr" rid="j_forma-2023-0005_ref_006">6</xref>].</p>
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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].

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

Formalized Mathematics

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In this article, we formalize the Gram-Schmidt process in the Mizar system [2], [3] (compare another formalization using Isabelle/HOL proof assistant [1])....

向量集正交化的Gram-Schmidt过程的形式化

在线发布:2023年9月26日

页面范围： 53 - 57

认可的:2023年3月31日

内政部： https://doi.org/10.2478/forma-2023-0005

关键词
Gram-Schmidt工艺,正交基,线性代数

©2023 Hiroyuki Okazaki，Sciendo出版

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