# 统计代写|主成分分析代写Principal Component Analysis代考|”Understanding Vectors, Matrix Multiplication, and Definiteness in Linear Algebra”

Vectors can be thought of as matrices with just one element in either row or column dimension, thus allowing for seamless extensions of the concepts of transpose and matrix multiplication to include vectors. By convention, vectors are represented as columns, meaning that the product Ax or x^T A might be meaningful, but x A or Ax^T wouldn’t unless the dimensions are both 1. However, certain computer systems may not enforce this strict distinction. The dot product or inner product of vectors x and y is denoted by x^Ty, consistent with this framework.

Matrix/Vector Product as a Linear Combination: The product of an n × m matrix A and a vector x can be expressed as a linear combination of the columns of A: Ax=∑i=1mxiaiAx = \sum_{i=1}^{m} x_i a_iAx=∑ i=1 m ​ x i ​ a i ​

where xi is a scalar and ai is a vector. Given the equation Ax = b, the vector b resides in the column space of A, spanned by the columns of A.

Outer Products: The outer product of vectors x and y results in a matrix: xyTxy^Txy T

Notably, the outer product doesn’t require the vectors to be of equal lengths, and unlike the inner product, it’s not commutative. A typical outer product is that of a vector with itself, xxTxx^Txx T , which generates a symmetric matrix.

Bilinear and Quadratic Forms, and Definiteness: The expression x^TAy is a bilinear form, and when A is a symmetric matrix, x^TAx is a quadratic form. While the quadratic form doesn’t necessitate A to be symmetric, we often work with symmetric matrices. The trace of the product plays a key role in handling quadratic forms due to its invariance under permutations, yielding the equality: xTAx=tr(xTAx)=tr(AxxT)x^TAx = \text{tr}(x^TAx) = \text{tr}(Axx^T)x T Ax=tr(x T Ax)=tr(Axx T )

Nonnegative Definite and Positive Definite Matrices: A symmetric matrix A is nonnegative definite if for every vector x, the quadratic form x^TAx is nonnegative (x^TAx ≥ 0). A matrix satisfying this condition is denoted by A ≽ 0. A symmetric matrix A is positive definite if for any nonzero vector x, x^TAx is strictly positive (x^TAx > 0); such a matrix is denoted by A ≻ 0. Nonnegative and positive definite matrices are crucial in many applications.

Anisometric Spaces: In cases where the properties derived from the standard inner product don’t adequately reflect the characteristics of a space due to differing scales or orientations, using a bilinear form x^TAy with a suitable matrix A allows for a more accurate representation. For instance, a diagonal matrix D reflecting the relative scales of axes could replace the identity matrix in the inner product.

Orthogonality with Respect to A: Two vectors x and y are said to be A-conjugate if they satisfy x^TAy = 0.

Elliptic Norms and Metrics: Generalizing the L2 vector norm, an elliptic norm associated with a symmetric positive definite matrix A is defined as: ∣∣x∣∣A=xTAx||x||_A = \sqrt{x^TAx}∣∣x∣∣ A ​ = x T Ax ​

This norm gives rise to an elliptic metric, where the distance between vectors x and y with respect to A is (x−y)TA(x−y)\sqrt{(x-y)^TA(x-y)} (x−y) T A(x−y) ​ . A significant metric in statistics is the Mahalanobis distance, which uses a covariance matrix S to measure distance: dM(x,y)=(x−y)TS−1(x−y)d_M(x, y) = (x – y)^T S^{-1} (x – y)d M ​ (x,y)=(x−y) T S −1 (x−y)

Mahalanobis distance provides a scale-aware metric that accounts for correlations among variables, making it especially useful in statistical 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。