施密特正交化的几何构建与代数刻画

Geometric Construction and Algebraic Characterization of Schmidt Orthogonalization

本文从几何空间中向量正交化的构造出发,分析了施密特正交化方法的几何构建过程以及推广到高维上的基本思路和底层逻辑,深化了对其几何意义的认识;借助齐次线性方程组解的理论和正交的代数表示,从代数的角度全新推导出其正交化公式,展示了几何问题代数化处理的转化思想.同时也讨论了一些非施密特正交化方法.

Starting from the construction of vectors orthogonalization in the geometric space,this paper reviews the geometric construction process of Schmidt orthogonalization method and the basic idea and underlying logic extended to higher dimensions,and deepens the understanding of its geometric significance.With the help of the solutions of homogeneous linear equations and orthogonal algebraic representation,the orthogonalization formula is derived from the perspective of algebra,which shows the transformation idea of algebraic treatment of geometric problems.Some none Schmidt orthogonalization methods are also discussed.