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数学代写|线性代数代写Linear algebra代考|MA2210 Characteristic and minimal polynomials

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数学代写|线性代数代写Linear algebra代考|Characteristic and minimal polynomials

We defined the determinant of a square matrix $A$. Now we want to define the determinant of a linear map $\alpha$. The obvious way to do this is to take the determinant of any matrix representing $\alpha$. For this to be a good definition, we need to show that it doesn’t matter which matrix we take; in other words, that $\operatorname{det}\left(A^{\prime}\right)=\operatorname{det}(A)$ if $A$ and $A^{\prime}$ are similar. But, if $A^{\prime}=P^{-1} A P$, then
$$\operatorname{det}\left(P^{-1} A P\right)=\operatorname{det}\left(P^{-1}\right) \operatorname{det}(A) \operatorname{det}(P)=\operatorname{det}(A),$$
since $\operatorname{det}\left(P^{-1}\right) \operatorname{det}(P)=1$. So our plan will succeed:
Definition $4.5$ (a) The determinant $\operatorname{det}(\alpha)$ of a linear map $\alpha: V \rightarrow V$ is the determinant of any matrix representing $T$.
(b) The characteristic polynomial $c_\alpha(x)$ of a linear map $\alpha: V \rightarrow V$ is the characteristic polynomial of any matrix representing $\alpha$.
(c) The minimal polynomial $m_\alpha(x)$ of a linear map $\alpha: V \rightarrow V$ is the monic polynomial of smallest degree which is satisfied by $\alpha$.

The second part of the definition is $\mathrm{OK}$, by the same reasoning as the first (since $c_A(x)$ is just a determinant). But the third part also creates a bit of a problem: how do we know that $\alpha$ satisfies any polynomial? The Cayley-Hamilton Theorem tells us that $c_A(A)=O$ for any matrix $A$ representing $\alpha$. Now $c_A(A)$ represents $c_A(\alpha)$, and $c_A=c_\alpha$ by definition; so $c_\alpha(\alpha)=O$. Indeed, the Cayley-Hamilton Theorem can be stated in the following form:

Proposition 4.7 For any linear map $\alpha$ on $V$, its minimal polynomial $m_\alpha(x)$ divides its characteristic polynomial $c_\alpha(x)$ (as polynomials).

Proof Suppose not; then we can divide $c_\alpha(x)$ by $m_\alpha(x)$, getting a quotient $q(x)$ and non-zero remainder $r(x)$; that is,
$$c_\alpha(x)=m_\alpha(x) q(x)+r(x) .$$
Substituting $\alpha$ for $x$, using the fact that $c_\alpha(\alpha)=m_\alpha(\alpha)=O$, we find that $r(\alpha)=$ 0 . But the degree of $r$ is less than the degree of $m_\alpha$, so this contradicts the definition of $m_\alpha$ as the polynomial of least degree satisfied by $\alpha$.

数学代写|线性代数代写Linear algebra代考|Jordan form

We finish this chapter by stating without proof a canonical form for matrices over the complex numbers under similarity.
Definition 4.6
(a) A Jordan block $J(n, \lambda)$ is a matrix of the form
$$\left[\begin{array}{ccccc} \lambda & 1 & 0 & \cdots & 0 \ 0 & \lambda & 1 & \cdots & 0 \ & & \cdots & & \ 0 & 0 & 0 & \cdots & \lambda \end{array}\right]$$
that is, it is an $n \times n$ matrix with $\lambda$ on the main diagonal, 1 in positions immediately above the main diagonal, and 0 elsewhere. (We take $J(1, \lambda)$ to be the $1 \times 1$ matrix $[\lambda]$.)
(b) A matrix is in Jordan form if it can be written in block form with Jordan blocks on the diagonal and zeros elsewhere.

Theorem 4.11 Over $\mathbb{C}$, any matrix is similar to a matrix in Jordan form; that is, any linear map can be represented by a matrix in Jordan form relative to a suitable basis. Moreover, the Jordan form of a matrix or linear map is unique apart from putting the Jordan blocks in a different order on the diagonal.

数学代写|线性代数代写Linear algebra代考|Characteristic and minimal polynomials

$$\operatorname{det}\left(P^{-1} A P\right)=\operatorname{det}\left(P^{-1}\right) \operatorname{det}(A) \operatorname{det}(P)=\operatorname{det}(A),$$

(b) 特征多项式 $c_\alpha(x)$ 线性映射 $\alpha: V \rightarrow V$ 是任何表示的矩阵的特征项式 $\alpha$.
(c) 最小多项式 $m_\alpha(x)$ 线性映射 $\alpha: V \rightarrow V$ 是满足以下条件的最小次数的一元多项式 $\alpha$.

$$c_\alpha(x)=m_\alpha(x) q(x)+r(x) .$$

数学代写|线性代数代写Linear algebra代考|Jordan form

(a) Jordan 块 $J(n, \lambda)$ 是形式的矩阵
$$\left[\begin{array}{llllllllllllllll} \lambda & 1 & 0 & \cdots & 0 & 0 & \lambda & 1 & \cdots & 0 & \cdots & 0 & 0 & 0 & \cdots & \lambda \end{array}\right]$$

(b) 如果矩阵可以写成块形式，且 Jordan 块在对角线上，其他地方为零，则该矩阵是 Jordan 形式的。

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。