# 统计代写|主成分分析代写Principal Component Analysis代考|”Understanding Norms, Metrics, and Orthogonality in Vector Spaces”

Norms Induced by Inner Products

Given an inner product space, the norm associated with an inner product $\langle \cdot, \cdot \rangle$ can be defined as the square root of the inner product of a vector with itself: $\|x\| = \sqrt{\langle x, x \rangle}$. This induced norm automatically satisfies the properties of a norm, including the triangle inequality as shown in equation (2.19). When applied to vectors, this induced norm coincides with the L2 norm (Euclidean norm). Normalized Vectors

Dividing a vector x by its Euclidean norm results in a normalized vector $\tilde{x} = \frac{x}{\|x\|}$ which has unit length (length 1). Metrics and Distances

A metric $\Delta(x, y)$ is a function that assigns a distance between any two elements in a set S. It should satisfy positivity, symmetry, and the triangle inequality. Given a norm $\|\cdot\|$, the metric $\Delta(x, y) = \|x – y\|$ is a natural way to define a distance between vectors. Orthogonal Vectors and Orthogonal Vector Spaces

Two vectors v1 and v2 are orthogonal if their inner product is zero ($\langle v1, v2 \rangle = 0$) and are denoted v1 ⊥ v2. A set of mutually orthogonal vectors, each of unit length, constitutes an orthonormal set. Two vector subspaces V1 and V2 of a larger vector space V are orthogonal (V1 ⊥ V2) if every vector in one subspace is orthogonal to every vector in the other. The orthogonal complement V2 = V1^\perp contains all vectors in V that are orthogonal to every vector in V1, and when V1 and V2 are both subspaces of V with V1 ⊕ V2 = V and dim(V1) + dim(V2) = dim(V), V2 is the orthogonal complement of V1. The One Vector and Means

The “one vector” or “summing vector” is a vector with all elements equal to 1, denoted by 1 or 1n. The arithmetic mean of the elements of an n-vector x is a scalar denoted by $\bar{x}$ and computed as $\bar{x} = \frac{1_n^Tx}{n}$, where 1_n is the one vector and Tx denotes the dot product of the one vector with x. The mean vector is an n-vector with all elements equal to the arithmetic mean, $\bar{x}$. In summary, inner products lead naturally to norms which, in turn, can define metrics or distances between vectors. Normalizing vectors creates unit vectors, and orthogonal vectors and subspaces play a significant role in understanding the structure of vector spaces. The one vector is a useful tool for computing sums and means of vector elements. The mean vector, being a constant multiple of the one vector, is also pertinent in several contexts.

### 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。