Jordan Canonical Form

Jordan 标准型 Jordan Canonical Form

定义 Definition

Jordan 标准型 是一种矩阵分解形式，每个方阵都可以表示为一个对角矩阵被若干 Jordan 块取代后的形式。这种形式使得处理矩阵的幂和矩阵函数更加简单。

A Jordan canonical form is a type of matrix decomposition where any square matrix can be represented in a form that replaces the diagonal matrix with several Jordan blocks. This form simplifies the handling of matrix powers and matrix functions.

Jordan 块 Jordan Block

一个 $n\times n$ 的 Jordan 块 ${\mathbf{J}}_n(\lambda )$ 是一个以下形式的矩阵：

An $n\times n$ Jordan block ${\mathbf{J}}_n(\lambda )$ is a matrix of the following form:

$${\mathbf{J}}_n(\lambda )=\left(\begin{array}{ccccc}\lambda & 1& 0& \cdots & 0\\ 0& \lambda & 1& \cdots & 0\\ 0& 0& \lambda & \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & 1\\ 0& 0& 0& \cdots & \lambda \end{array}\right)$$

其中，$\lambda$ 是特征值。

where $\lambda$ is an eigenvalue.

性质 Properties

1. 特征值 Eigenvalues：Jordan 标准型的对角线元素都是原矩阵的特征值。

   The diagonal elements of the Jordan canonical form are the eigenvalues of the original matrix.

2. 相似矩阵 Similarity：每个矩阵 $\mathbf{A}$ 都与它的 Jordan 标准型 $\mathbf{J}$ 相似，即存在可逆矩阵 $\mathbf{P}$ 使得 $\mathbf{A}={\mathbf{PJP}}^{-1}$。

   Every matrix $\mathbf{A}$ is similar to its Jordan canonical form $\mathbf{J}$, meaning there exists an invertible matrix $\mathbf{P}$ such that $\mathbf{A}={\mathbf{PJP}}^{-1}$.

3. 唯一性 Uniqueness：Jordan 标准型在相似矩阵的意义上是唯一的。

   The Jordan canonical form is unique up to similarity transformations.

4. 块对角形式 Block Diagonal Form：Jordan 标准型由若干 Jordan 块构成，其大小和数量由矩阵 $\mathbf{A}$ 的特征值和特征向量决定。

   The Jordan canonical form is composed of several Jordan blocks, the size and number of which are determined by the eigenvalues and eigenvectors of the matrix $\mathbf{A}$.

证明 Proof

1. 特征值分解 Eigenvalue Decomposition：首先，对矩阵 $\mathbf{A}$ 进行特征值分解，找出所有特征值和对应的特征向量。

   First, perform eigenvalue decomposition on the matrix $\mathbf{A}$ to find all eigenvalues and their corresponding eigenvectors.

2. 构建 Jordan 块 Construct Jordan Blocks：根据特征值和特征向量，构建对应的 Jordan 块。对于每个特征值 ${\lambda }_i$，若其代数重数为 $m$，几何重数为 $g$，则需要构造 $m-g$ 个大小大于 $1$ 的 Jordan 块。

   Construct the corresponding Jordan blocks based on the eigenvalues and eigenvectors. For each eigenvalue ${\lambda }_i$, if its algebraic multiplicity is $m$ and its geometric multiplicity is $g$, construct $m-g$ Jordan blocks of size greater than 1.

   代数重数 (Algebraic Multiplicity): 特征值在特征多项式中作为根出现的次数。
   The number of times a eigenvalue appears as a root of the characteristic polynomial.

   几何重数 (Geometric Multiplicity): 与特征值相关的线性独立特征向量的个数，即零空间的维数。
   The number of linearly independent eigenvectors associated with an eigenvalue, equivalent to the dimension of the null space.

3. 构造变换矩阵 Construct the Transformation Matrix：构造变换矩阵 $\mathbf{P}$，其列向量由特征向量和广义特征向量组成。

   Construct the transformation matrix $\mathbf{P}$, whose column vectors consist of the eigenvectors and generalized eigenvectors.

4. 验证 Verification：验证 ${\mathbf{P}}^{-1}\mathbf{AP}$ 是否为 Jordan 标准型。若是，则 $\mathbf{J}={\mathbf{P}}^{-1}\mathbf{AP}$。

   Verify that ${\mathbf{P}}^{-1}\mathbf{AP}$ is in Jordan canonical form. If so, then $\mathbf{J}={\mathbf{P}}^{-1}\mathbf{AP}$.

例子 Example

设 $\mathbf{A}$ 为以下矩阵：

Let $\mathbf{A}$ be the following matrix:

$$\mathbf{A}=\left(\begin{array}{cccc}5& 4& 2& 1\\ 0& 1& -1& -1\\ 0& 0& 2& 1\\ 0& 0& 0& 3\end{array}\right)$$

其 Jordan 标准型为：

Its Jordan canonical form is:

$$\mathbf{J}=\left(\begin{array}{cccc}5& 1& 0& 0\\ 0& 5& 0& 0\\ 0& 0& 2& 1\\ 0& 0& 0& 3\end{array}\right)$$