# 统计代写|主成分分析代写Principal Component Analysis代考|”Understanding Matrix Rank: Linear Dependence, Full Rank, and Rank Deficiency”

Matrix rank plays a significant role in understanding the linear dependence or independence of the rows or columns of a matrix. The rank of a matrix AAA, denoted as rank(A)\text{rank}(A)rank(A), represents the maximum number of linearly independent rows or columns. It’s essential to note that scaling a matrix by a non-zero scalar doesn’t affect its rank, hence rank(aA)=rank(A)\text{rank}(aA) = \text{rank}(A)rank(aA)=rank(A) for a≠0a \neq 0a  =0. According to Section 2.1, the rank of an n×mn \times mn×m matrix AAA is always less than or equal to the minimum of its dimensions: rank(A)≤min⁡(n,m).\text{rank}(A) \leq \min(n, m).rank(A)≤min(n,m).

Row rank and column rank are equivalent; thus, referring simply to “rank” suffices. Assume matrix AAA has ppp linearly independent rows and qqq linearly independent columns. By permuting rows and columns, it can be rearranged so that the first ppp rows and qqq columns form a linearly independent set. It can be shown that ppp must equal qqq, proving that the maximum number of linearly independent rows equals the maximum number of linearly independent columns.

Therefore, we have:

rank(A)=dim⁡(V(A))\text{rank}(A) = \dim(V(A))rank(A)=dim(V(A)) rank(A⊤)=rank(A)\text{rank}(A^\top) = \text{rank}(A)rank(A ⊤ )=rank(A) dim⁡(V(A⊤))=dim⁡(V(A))\dim(V(A^\top)) = \dim(V(A))dim(V(A ⊤ ))=dim(V(A)) A matrix is said to be full rank if its rank equals its smaller dimension. Otherwise, it is rank deficient, and its rank deficiency is the difference between its smaller dimension and its rank. A square full rank matrix is called nonsingular (invertible), while a singular matrix is not invertible.

Square matrices that are either row or column diagonally dominant are nonsingular. Positive definite matrices are also nonsingular.

Elementary matrices, due to their structure, have full rank. Furthermore, the rank of a matrix after multiplication by an elementary matrix remains unchanged. Consequently, for any product of elementary matrices PPP and QQQ with a given matrix AAA, the following holds: rank(PAQ)=rank(A)\text{rank}(PAQ) = \text{rank}(A)rank(PAQ)=rank(A).

When considering partitioned matrices, the rank of a matrix AAA partitioned into blocks AijA_{ij}A ij ​ satisfies: rank(Aij)≤rank(A)\text{rank}(A_{ij}) \leq \text{rank}(A)rank(A ij ​ )≤rank(A).

Moreover, rank inequalities arise when examining sums and products of matrices:

The rank of AAA is bounded by the sum of ranks of its block-row partitions: rank(A)≤rank([A11∣A12])+rank([A21∣A22])\text{rank}(A) \leq \text{rank}([A_{11}|A_{12}]) + \text{rank}([A_{21}|A_{22}])rank(A)≤rank([A 11 ​ ∣A 12 ​ ])+rank([A 21 ​ ∣A 22 ​ ]). Similar bounds exist for block-column partitions. Under certain conditions of orthogonality among the row spaces of the submatrices, the rank of AAA can be decomposed as the sum of ranks of its block partitions. This is particularly true when the row spaces of the submatrices are orthogonal complements of each other.

### 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。