# 统计代写|主成分分析代写Principal Component Analysis代考|”Elementary Permutation Matrices and Transformations in Matrix Computations”

In matrix computations, interchanging rows or columns is a common operation that can lead to more insightful matrix partitions or facilitate certain computations. Instead of physically moving data, these permutations are achieved by changing indexing, using elementary permutation matrices.

An elementary permutation matrix is derived from the identity matrix by swapping two rows (or columns). For instance, to interchange rows p and q in a matrix A, one premultiplies A by the elementary permutation matrix Epq, where Epq exchanges the pth and qth rows of the identity matrix. Notably, Epq = Eqp and EpqEpq = I, indicating that every elementary permutation matrix is its own inverse.

Successive premultiplications by appropriate elementary permutation matrices can realize any permutation of the rows of a matrix, and the total number of permutations required to achieve a specific reordering is always even or odd. Moreover, any given permutation can be attained by a series of adjacent row swaps.

Postmultiplying a matrix A by Epq results in the interchange of the pth and qth columns. Permutations of both rows and columns can be represented as B = Eπ1AEπ2, where Eπ1 and Eπ2 are permutation matrices for row and column permutations respectively.

Another transformation frequently used is scalar multiplication of rows or columns to normalize their norms or for other computational advantages. This is done via multiplication by an elementary transformation matrix, denoted by Ep(a), which is the identity matrix with the pth diagonal entry replaced by the scalar ‘a’. For row multiplication, Ep(a) is used for premultiplication, while for column multiplication, it is postmultiplied, with its size being equal to the number of columns of the target matrix. The inverse operation is simply a multiplication by the reciprocal scalar using the same row, i.e., A = Ep(1/a)Ep(a)A.

Additionally, there exists a special permutation matrix, called the vec-permutation matrix Knm, which transforms vec(A) into vec(AT) when multiplied by it. This matrix has dimensions nm × nm, where A is an n × m matrix.

### 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。