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If $P$ is a stochastic matrix, then a steady-state vector (or equilibrium vector) for $P$ is a probability vector $\mathbf{q}$ such that
$$P \mathbf{q}=\mathbf{q}$$
It can be shown that every stochastic matrix has a steady-state vector. In Example 3, $\mathbf{q}$ is a steady-state vector for $P$.

EXAMPLE 4 The probability vector $\mathbf{q}=\left[\begin{array}{l}.375 \ .625\end{array}\right]$ is a steady-state vector for the population migration matrix $M$ in Example 1, because
$$M \mathbf{q}=\left[\begin{array}{ll} .95 & .03 \ .05 & .97 \end{array}\right]\left[\begin{array}{l} .375 \ .625 \end{array}\right]=\left[\begin{array}{l} .35625+.01875 \ .01875+.60625 \end{array}\right]=\left[\begin{array}{l} .375 \ .625 \end{array}\right]=\mathbf{q}$$

If the total population of the metropolitan region in Example 1 is 1 million, then q from Example 4 would correspond to having 375,000 persons in the city and 625,000 in the suburbs. At the end of one year, the migration out of the city would be $(.05)(375,000)=18,750$ persons, and the migration into the city from the suburbs would be $(.03)(625,000)=18,750$ persons. As a result, the population in the city would remain the same. Similarly, the suburban population would be stable.
The next example shows how to find a steady-state vector.

## 数学代写|线性代数代写Linear algebra代考|EIGENVECTORS AND EIGENVALUES

Although a transformation $\mathbf{x} \mapsto A \mathbf{x}$ may move vectors in a variety of directions, it often happens that there are special vectors on which the action of $A$ is quite simple.

EXAMPLE 1 Let $A=\left[\begin{array}{rr}3 & -2 \ 1 & 0\end{array}\right], \mathbf{u}=\left[\begin{array}{r}-1 \ 1\end{array}\right]$, and $\mathbf{v}=\left[\begin{array}{l}2 \ 1\end{array}\right]$. The images of $\mathbf{u}$ and $\mathbf{v}$ under multiplication by $A$ are shown in Figure 1. In fact, $A \mathbf{v}$ is just $2 \mathbf{v}$. So $A$ only “stretches,” or dilates, $\mathbf{v}$.

As another example, readers of Section $4.9$ will recall that if $A$ is a stochastic matrix, then the steady-state vector $\mathbf{q}$ for $A$ satisfies the equation $A \mathbf{x}=\mathbf{x}$. That is, $A \mathbf{q}=1 \cdot \mathbf{q}$.

This section studies equations such as
$$A \mathbf{x}=2 \mathbf{x} \quad \text { or } \quad A \mathbf{x}=-4 \mathbf{x}$$
where special vectors are transformed by $A$ into scalar multiples of themselves.
An eigenvector of an $n \times n$ matrix $A$ is a nonzero vector $\mathbf{x}$ such that $A \mathbf{x}=\lambda \mathbf{x}$ for some scalar $\lambda$. A scalar $\lambda$ is called an eigenvalue of $A$ if there is a nontrivial solution $\mathbf{x}$ of $A \mathbf{x}=\lambda \mathbf{x}$; such an $\mathbf{x}$ is called an eigenvector corresponding to $\lambda .{ }^1$
It is easy to determine if a given vector is an eigenvector of a matrix. It is also easy to decide if a specified scalar is an eigenvalue.

## 数学代写|线性代数代写线性代数代考|稳态向量

$$P \mathbf{q}=\mathbf{q}$$

$$M \mathbf{q}=\left[\begin{array}{ll} .95 & .03 \ .05 & .97 \end{array}\right]\left[\begin{array}{l} .375 \ .625 \end{array}\right]=\left[\begin{array}{l} .35625+.01875 \ .01875+.60625 \end{array}\right]=\left[\begin{array}{l} .375 \ .625 \end{array}\right]=\mathbf{q}$$

## 数学代写|线性代数代写线性代数代考|特征向量和特征值

EXAMPLE 1让$A=\left[\begin{array}{rr}3 & -2 \ 1 & 0\end{array}\right], \mathbf{u}=\left[\begin{array}{r}-1 \ 1\end{array}\right]$，和$\mathbf{v}=\left[\begin{array}{l}2 \ 1\end{array}\right]$。$\mathbf{u}$和$\mathbf{v}$在乘以$A$后的图像如图1所示。实际上，$A \mathbf{v}$就是$2 \mathbf{v}$。所以$A$只“拉伸”或扩张$\mathbf{v}$ .

$$A \mathbf{x}=2 \mathbf{x} \quad \text { or } \quad A \mathbf{x}=-4 \mathbf{x}$$

## 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。