Matrix Theory

矩阵理论 Matrix Theory

Cayley-Hamilton 定理 Cayley-Hamilton Theorem

定义 Definition

Cayley-Hamilton 定理: 一个 $n\times n$ 的矩阵 $A$ 满足其特征多项式 $p(\lambda )$，即 $A$ 代入其特征多项式得到的矩阵多项式等于零矩阵 Cayley-Hamilton Theorem: An $n\times n$ matrix $A$ satisfies its characteristic polynomial $p(\lambda )$, meaning that the matrix polynomial obtained by substituting $A$ into its characteristic polynomial equals the zero matrix

用数学语言表达:

Expressed mathematically:

$$p(\lambda )=\det (\lambda I-A)$$ $$p(A)=0$$

特征多项式 Characteristic Polynomial

一个矩阵 $A$ 的特征多项式定义为:

The characteristic polynomial of a matrix $A$ is defined as:

$$p(\lambda )=\det (\lambda I-A)$$

其中 $\det$ 表示行列式，$I$ 是单位矩阵，$\lambda$ 是一个标量

where $\det$ denotes the determinant, $I$ is the identity matrix, and $\lambda$ is a scalar

证明 Proof

Cayley-Hamilton 定理的证明通常包括以下步骤:

The proof of the Cayley-Hamilton Theorem typically includes the following steps:

1. 计算 $A$ 的特征多项式 $p(\lambda )$ Calculate the characteristic polynomial $p(\lambda )$ of $A$
2. 通过代入 $\lambda =A$ 验证 $p(A)=0$ Verify that $p(A)=0$ by substituting $\lambda =A$

性质 Properties

1. 适用于任何方阵: Cayley-Hamilton 定理对任何 $n\times n$ 矩阵都适用，无论是实矩阵还是复矩阵 Applicable to any square matrix: The Cayley-Hamilton Theorem applies to any $n\times n$ matrix, whether real or complex
2. 特征值和特征向量的关系: Cayley-Hamilton 定理表明矩阵的特征值是其特征多项式的根 Relationship between eigenvalues and eigenvectors: The Cayley-Hamilton Theorem indicates that the eigenvalues of a matrix are the roots of its characteristic polynomial

应用 Applications

1. 求幂矩阵: 利用 Cayley-Hamilton 定理可以简化高次幂矩阵的计算 Computing matrix powers: The Cayley-Hamilton Theorem can simplify the computation of high powers of a matrix
2. 求矩阵函数: 如矩阵的指数函数、对数函数等 Finding matrix functions: Such as the exponential function and logarithm function of a matrix
3. 系统理论: 在控制系统的稳定性分析中 Systems theory: In the stability analysis of control systems

计算技巧 Calculation Techniques

• 利用幂简化: 通过 Cayley-Hamilton 定理，可以将高次幂矩阵 $A^k$ 表示为较低次幂矩阵的线性组合，从而简化计算 Simplifying powers: By using the Cayley-Hamilton Theorem, a high power of a matrix $A^k$ can be expressed as a linear combination of lower powers of the matrix, thus simplifying the calculation
• 求解线性微分方程组: 在求解常系数线性微分方程组时，可以利用 Cayley-Hamilton 定理将问题转化为代数问题 Solving linear differential equations: When solving constant coefficient linear differential equations, the Cayley-Hamilton Theorem can be used to transform the problem into an algebraic one

易混淆点 Easily Confused Points

• 特征多项式与最小多项式的区别: 特征多项式是矩阵的特征值的多项式，最小多项式是满足矩阵等式 $p(A)=0$ 的最低次多项式 Difference between characteristic polynomial and minimal polynomial: The characteristic polynomial is the polynomial of the eigenvalues of the matrix, while the minimal polynomial is the lowest-degree polynomial that satisfies the matrix equation $p(A)=0$
• 矩阵与其特征值: Cayley-Hamilton 定理表明矩阵的特征多项式应用于矩阵本身等于零矩阵，但不意味着矩阵本身为零 Matrix and its eigenvalues: The Cayley-Hamilton Theorem indicates that the characteristic polynomial of a matrix, when applied to the matrix itself, equals the zero matrix, but this does not mean the matrix itself is zero

示例 Examples

示例 1: 计算矩阵的特征多项式并验证 Cayley-Hamilton 定理

Example 1: Calculate the characteristic polynomial of a matrix and verify the Cayley-Hamilton Theorem

设 $A$ 为一个 $2\times 2$ 矩阵:

Let $A$ be a $2\times 2$ matrix:

$$A=\left(\begin{array}{cc}4& 1\\ 2& 3\end{array}\right)$$

1. 计算特征多项式: Calculate the characteristic polynomial:

$$p(\lambda )=\det (\lambda I-A)=\det \left(\begin{array}{cc}\lambda -4& -1\\ -2& \lambda -3\end{array}\right)=(\lambda -4)(\lambda -3)-(-2)(-1)={\lambda }^2-7\lambda +10$$

1. 验证 $p(A)=0$: Verify $p(A)=0$:

$$p(A)=A^2-7A+10I$$

计算 $A^2$:

Calculate $A^2$:

$$A^2=\left(\begin{array}{cc}4& 1\\ 2& 3\end{array}\right)\left(\begin{array}{cc}4& 1\\ 2& 3\end{array}\right)=\left(\begin{array}{cc}18& 7\\ 14& 11\end{array}\right)$$

然后:

Then:

$$p(A)=\left(\begin{array}{cc}18& 7\\ 14& 11\end{array}\right)-7\left(\begin{array}{cc}4& 1\\ 2& 3\end{array}\right)+10\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$$

这验证了 Cayley-Hamilton 定理

This verifies the Cayley-Hamilton Theorem

示例 2: 利用 Cayley-Hamilton 定理计算高次幂矩阵

Example 2: Using the Cayley-Hamilton Theorem to compute higher powers of a matrix

对于矩阵 $A$:

For the matrix $A$:

$$A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$$

特征多项式为:

The characteristic polynomial is:

$$p(\lambda )={\lambda }^2-5\lambda -2$$

根据 Cayley-Hamilton 定理:

According to the Cayley-Hamilton Theorem:

$$A^2-5A+2I=0⟹A^2=5A-2I$$

因此可以简化 $A^3$ 的计算:

Thus, the computation of $A^3$ can be simplified:

$$A^3=A\cdot A^2=A(5A-2I)=5A^2-2A=5(5A-2I)-2A=25A-10I-2A=23A-10I$$

3x3 矩阵的 Cayley-Hamilton 定理 Cayley-Hamilton Theorem for a 3x3 Matrix

对于一个 $3\times 3$ 矩阵 $A$:

For a $3\times 3$ matrix $A$:

$$A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$$

其特征多项式可以表示为:

Its characteristic polynomial can be expressed as:

$$p(\lambda )=\det (\lambda I-A)={\lambda }^3-\text{tr}(A){\lambda }^2+\text{sum of principal minors of }A\cdot \lambda -\det (A)$$

其中:

Where:

• $\text{tr}(A)$ 表示矩阵 $A$ 的迹，即对角线上元素的和:

  $\text{tr}(A)$ denotes the trace of the matrix $A$, which is the sum of the diagonal elements:

  $$\text{tr}(A)=a_{11}+a_{22}+a_{33}$$

$

$

• 矩阵 $A$ 的所有主子式之和 The sum of the principal minors of matrix $A$

例如，对于矩阵 $A$:

For example, for the matrix $A$:

$$A=\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$$

其所有主子式的行列式之和为:

The sum of the determinants of all its principal minors is:

$$\text{sum of principal minors}=\det \left(\begin{array}{cc}a& b\\ d& e\end{array}\right)+\det \left(\begin{array}{cc}a& c\\ g& i\end{array}\right)+\det \left(\begin{array}{cc}e& f\\ h& i\end{array}\right)$$

应用 Cayley-Hamilton 定理，将特征多项式 $p(\lambda )$ 应用于矩阵 $A$，得到:

Applying the Cayley-Hamilton Theorem, the characteristic polynomial $p(\lambda )$ is applied to the matrix $A$ to obtain:

$$p(A)=A^3-\text{tr}(A)A^2+\text{sum of principal minors of }A\cdot A-\det (A)I=0$$