统计代写|主成分分析代写Principal Component Analysis代考|”Key Concepts and Applications of Matrices in Data Sciences”

In this chapter, the focus is on matrices with real elements, though some properties extend to complex matrices. Key concepts and terminology are presented, motivated by their applications in data sciences.

Vector Spaces: Rows and columns of a matrix can be treated as vectors. The column space (V(A) or span(A)) is the vector space generated by the columns of an n × m matrix A, having dimension m or less and order n. Similarly, the row space is the vector space generated by the rows, having dimension n or less and order m.

Matrix Multiplication by Scalar: Similar to vector-scalar multiplication, a matrix A multiplied by a scalar c is defined as cA, where each element aij is multiplied by c.

Matrix Structure: Elements aij on the principal diagonal are called diagonal elements. Elements above and below the diagonal are termed accordingly. A matrix is symmetric if aij = aji, skew-symmetric if aij = -aji, and Hermitian if aij = ¯aji (with ¯a being the conjugate of a complex number a). Hermitian matrices are self-adjoint and square.

Sparse Matrices: Many matrices have a high proportion of zero elements, known as sparse matrices. Diagonal matrices are a key example of sparse matrices, having non-zero elements only on the diagonal.

Matrix Forms: Various forms of matrices are introduced, including upper/lower triangular, trapezoidal, band, and Hessenberg matrices. Special patterns in matrices are often characterized by the location of zeros and non-zeros.

Matrix Shaping Operators:

Transpose: The transpose of a matrix A (AT) flips rows and columns. For complex matrices, the conjugate transpose (AH) takes the complex conjugate of each element and transposes the matrix. Diagonal Matrices: The diag() function constructs a diagonal matrix from a vector, placing its elements along the diagonal. Throughout the text, it is emphasized that many properties of matrices discussed apply to matrices with finite elements and dimensions, leading to finite-order vector spaces. Later chapters delve deeper into specific types of matrices relevant to regression analysis, multivariate data analysis, and statistical applications, taking advantage of their unique properties for efficient computations.

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。