# 统计代写|主成分分析代写Principal Component Analysis代考|”Orthogonalization, Orthonormal Basis, Vector Approximation, and Properties of Flats and Cones”

Gram-Schmidt Orthogonalization Process (Modified): The modified Gram-Schmidt algorithm sequentially orthogonalizes a set of linearly independent vectors x1,…,xmx_1, …, x_mx 1 ​ ,…,x m ​ in Rn\mathbb{R}^nR n to create an orthonormal set x~1,…,x~m\tilde{x}_1, …, \tilde{x}_m x ~

1 ​ ,…, x ~

m ​ . Each step ensures that x~k\tilde{x}_k x ~

k ​ is orthogonal to the previously created orthonormal vectors x~1,…,x~k−1\tilde{x}_1, …, \tilde{x}_{k-1} x ~

1 ​ ,…, x ~

k−1 ​ before normalization. This method reduces the impact of rounding errors compared to other approaches.

Orthonormal Basis Sets and Fourier Expansions: An orthonormal basis set u1,…,unu_1, …, u_nu 1 ​ ,…,u n ​ allows any vector xxx in the space to be represented as a linear combination x=c1u1+⋯+cnunx = c_1u_1 + \cdots + c_nu_nx=c 1 ​ u 1 ​ +⋯+c n ​ u n ​ , with the Fourier coefficients ci=⟨x,ui⟩c_i = \langle x, u_i \ranglec i ​ =⟨x,u i ​ ⟩. Parseval’s identity states that the squared norm of xxx is equal to the sum of the squared Fourier coefficients: ∣∣x∣∣2=∑i=1nci2||x||^2 = \sum_{i=1}^{n} c_i^2∣∣x∣∣ 2 =∑ i=1 n ​ c i 2 ​ , which provides a consistency across different choices of orthonormal bases.

Approximation of Vectors: When approximating a vector xxx in a higher-dimensional space by a vector x~\tilde{x} x ~ in a lower-dimensional subspace VVV, the optimal approximation under the Euclidean norm is achieved by retaining the basis vectors u1,…,uku_1, …, u_ku 1 ​ ,…,u k ​ with the largest absolute Fourier coefficients cic_ic i ​ .

Flats, Hyperplanes, and Linear Systems: Flats are sets of points defined by systems of linear equations, which can represent affine subspaces. When the equations are homogeneous, the flat goes through the origin and becomes a vector subspace. Flats of particular dimensions (hyperplanes and lines) are also discussed.

Cones and Their Properties: Convex cones are important in optimization problems. The dual cone V∗V^*V ∗ and polar cone V0V_0V 0 ​ are defined in relation to a given set of vectors VVV, with V0V_0V 0 ​ being the set of vectors that are non-positive with respect to all vectors in VVV. The union of the dual and polar cones of a convex cone forms a vector space.

Cross Product in R3\mathbb{R}^3R 3 : In three-dimensional space, the cross product x×yx \times yx×y of two vectors xxx and yyy is another vector with properties such as self-nilpotency (meaning x×x=0x \times x = 0x×x=0), anti-commutativity, and linearity with respect to scalar multiplication. It provides a perpendicular vector to both xxx and yyy with a magnitude representing the area of the parallelogram spanned by xxx and yyy. The cross product is not defined in higher dimensions unless generalized to exterior algebra or wedge products.

Throughout the text, it’s emphasized how the mathematical representation of concepts (like the Gram-Schmidt process or vector approximations) and their computational implementation can differ for stability and efficiency reasons. The text also highlights the importance of orthonormal basis sets in simplifying calculations and the utility of vector products like the cross product in R3\mathbb{R}^3R 3 for geometric interpretations and applications.

### MATLAB代写

MATLAB 是一款高性能的技术计算语言，集成了计算、可视化和编程环境于一体，以熟悉的数学符号表达问题和解决方案。MATLAB 的基本数据元素是一个不需要维度的数组，使得能够快速解决带有矩阵和向量公式的多种技术计算问题，相比使用 C 或 Fortran 等标量非交互式语言编写的程序，效率大大提高。MATLAB 名称源自“矩阵实验室”（Matrix Laboratory）。最初开发 MATLAB 的目标是为了提供对 LINPACK 和 EISPACK 项目的矩阵软件的便捷访问，这两个项目代表了当时矩阵计算软件的先进技术。经过长期发展和众多用户的贡献，MATLAB 已成为数学、工程和科学入门及高级课程的标准教学工具，在工业界，MATLAB 是高效研究、开发和分析的理想选择。MATLAB 提供了一系列名为工具箱的特定应用解决方案集，这对广大 MATLAB 用户至关重要，因为它们极大地扩展了 MATLAB 环境，使其能够解决特定类别问题。工具箱包含了针对特定应用领域的 MATLAB 函数（M 文件），涵盖信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等诸多领域。