# 统计代写|主成分分析代写Principal Component Analysis代考|”Key Concepts in Cartesian Geometry: Projections, Mean Vectors, and Orthogonalization Transformations”

Cartesian Geometry: In a d-dimensional space, a Cartesian coordinate system is defined by d orthonormal unit vectors eie_ie i ​ where i=1,…,di = 1, …, di=1,…,d. Any point xxx with coordinates (x1,…,xd)(x_1, …, x_d)(x 1 ​ ,…,x d ​ ) corresponds to a vector pointing from the origin to that point. This vector can be expressed as a linear combination of the unit vectors.

Projections: The projection of vector yyy onto vector xxx is given by y^=⟨x,y⟩∥x∥2x\hat{y} = \frac{\langle x, y \rangle}{\|x\|^2} x y ^ ​ = ∥x∥ 2

⟨x,y⟩ ​ x, where ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle⟨⋅,⋅⟩ represents the inner product. Geometrically, this projects yyy onto the line parallel to xxx and scales it according to the component of yyy along xxx. The residual vector r=y−y^r = y – \hat{y}r=y− y ^ ​ is orthogonal to y^\hat{y} y ^ ​ .

Mean Vector: The mean vector yˉ\bar{y} y ˉ ​ can be viewed as the projection of vector yyy onto the one vector 1n1_n1 n ​ , where yˉ=1nTyn⋅1n\bar{y} = \frac{1_n^Ty}{n} \cdot 1_n y ˉ ​ = n 1 n T ​ y ​ ⋅1 n ​ .

Angles Between Vectors: The angle θ\thetaθ between two non-zero vectors xxx and yyy is found by calculating the cosine of the angle using the lengths and inner product: θ=cos⁡−1(⟨x,y⟩∥x∥⋅∥y∥)\theta = \cos^{-1} \left( \frac{\langle x, y \rangle}{\|x\| \cdot \|y\|} \right)θ=cos −1 ( ∥x∥⋅∥y∥ ⟨x,y⟩ ​ ).

Direction Cosines: The direction cosines of a vector are the cosines of the angles it forms with the unit axes, and they are proportional to the scaled components of the vector.

High-Dimensional Geometries: As the dimensionality of the space increases, vectors tend to become nearly orthogonal due to the concentration of measure phenomenon, even when they are not intentionally constructed to be so.

Orthogonalization Transformations: The Gram-Schmidt process is introduced as a method to convert a set of linearly independent vectors x1,…,xmx_1, …, x_mx 1 ​ ,…,x m ​ into an orthonormal set x~1,…,x~m\tilde{x}_1, …, \tilde{x}_m x ~

1 ​ ,…, x ~

m ​ . This is done iteratively, normalizing each vector and projecting subsequent vectors orthogonally onto the subspace spanned by the previously processed orthonormal vectors.

The text also mentions that more advanced topics, such as projections onto higher-dimensional subspaces, linear regression as a projection, rotations, and further details about the Gram-Schmidt process, will be discussed later in the text.

### 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。