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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

电气工程代写|数字信号过程代写digital signal process代考|Eigenvectors, Eigenvalues and Matrix Decompositions

For some of the linear transformations that we have covered in the previous section, it may be possible to just look at their matrices and make an educated guess as to what they do. However, in most practical applications, the transformation matrix will not reveal its secrets so easily, and we need a more rigorous analysis approach that yields parameters indicating the transformation’s underlying mechanisms. One such analysis is the calculation of the eigenvalues and eigenvectors of a transformation matrix. These eigen-parameters are important in a variety of engineering disciplines including mechanical and chemical engineering, but they will also play a part in our later discussion on MIMO (multiple-input multiple-output) communication systems. After having found the eigenvalues and vectors, it is a simple matter to complete a mathematic process called the eigenvalue decomposition. The eigenvalue decomposition manages to break the original transformation matrix apart to produce three separate matrices, whose product yields the original transformation matrix. Beside the fact that we can immediately read off the eigenvalues and eigenvectors from the three matrices, the decomposition itself enables us to undertake certain mathematical operations that would have been difficult to do on the original transformation matrix. There are in fact several additional decompositions including the polar and singular value decompositions, and each allows us unique insight into the underlying mechanisms of the associated transformation.

电气工程代写|数字信号过程代写digital signal process代考|What are Eigenvalues and Eigenvectors

As the reader can readily imagine, there are actually an infinite number of possible vectors that we could draw both in the plus and minus $x$-axis sense whose direction would not be affected by the rotation. These vectors would simply differ in length. The collection of all these vectors is called the eigenspace, and their transformation scales each one of them by the eigenvalue, $\lambda_1=1$. The idea that an eigenvector, $v$, is a vector remaining unchanged in direction after a transformation via the matrix $A$ and scales in magnitude by the eigenvalue, $\lambda$, is expressed as follows.
$$A v=\lambda v$$
To find a solution for $v$, we will rewrite the equation where the identity matrix $I$ and $A$ have the same dimension.
$$\begin{gathered} A v=\lambda v \ (A-\lambda I) v=0 \end{gathered}$$
In section 1.2.2.3, under the subheading called ‘The Matrix Inversion’, we look back at rules 2 and 3 to realize that if $(A-\lambda I)$ is invertible, then $v$ must be the zero vector and as mentioned earlier, $\left[\begin{array}{lll}0 & 0 & . .\end{array}\right]$ is not a valid eigenvector. We must therefore choose the eigenvalues $(\lambda)$ such that $(A-\lambda I)$ cannot be inverted which means that its determinant must be zero.
$$\operatorname{det}(A-\lambda I)=0$$
The eigenvalues of $A$ are simply the roots of that characteristic polynomial, $p(\lambda)=\operatorname{det}(A-\lambda I)$. Now as we already know, if the determinant is zero, there will be either no or an infinite number of solutions. This supports our understanding that there may be an infinite number of eigenvectors in a particular eigenspace associated with one eigenvalue.

数字信号过程代考

电气工程代写|数字信号过程代写digital signal process代考|What are Eigenvalues and Eigenvectors

$$A v=\lambda v$$

$$A v=\lambda v(A-\lambda I) v=0$$

$$\operatorname{det}(A-\lambda I)=0$$

有限元方法代写

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MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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