若尔当标准形 弗罗贝尼乌斯标准形

小结：

1、相似矩阵提出动机、诞生背景：化繁为简。

AB对B施加线性变换A，但A较复杂---》変基，变B所在空间的基 ---》化简了A

https://en.wikipedia.org/wiki/Matrix_similarity

Similarity is an equivalence relation on the space of square matrices.

Because matrices are similar if and only if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator:

• Rank
• Characteristic polynomial, and attributes that can be derived from it:
  • Determinant
  • Trace
  • Eigenvalues, and their algebraic multiplicities
• Geometric multiplicities of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix P used).
• Minimal polynomial
• Frobenius normal form
• Jordan normal form, up to a permutation of the Jordan blocks
• Index of nilpotence
• Elementary divisors, which form a complete set of invariants for similarity of matrices over a principal ideal domain.
  

 

 

【“两个相似的矩阵可以看做是同一个线性变换的‘两面’，即在两个不同的基下的表现”：故特征向量不一定相同外，其他都相同，

其中“其他”指矩阵的其他描述量，如行列式的值、秩、迹数、特征值、特征多项式、初等因子；根据逆否命题的成立关系，可出

2个矩阵为相似矩阵的必要条件。】

 

https://zh.wikipedia.org/wiki/相似矩陣

 

两个相似的矩阵有许多相同的性质：

• 两者的秩相等。
• 两者的行列式值相等。
• 两者的迹数相等。
• 两者拥有同样的特征值，尽管相应的特征向量一般不同。
• 两者拥有同样的特征多项式。
• 两者拥有同样的初等因子。

这种现象的原因有两个：

• 两个相似的矩阵可以看做是同一个线性变换的“两面”，即在两个不同的基下的表现。
• 映射X  P⁻¹XP是从n阶方阵射到n阶方阵的一个双射同构，因为P是可逆的。

因此，在给定了矩阵A后，只要能找到一个与之相似而又足够“简单”的“规范形式”B，那么对A的研究就可以转化为对更简单的矩阵B的研究。比如说A被称为可对角化的，如果它与一个对角矩阵相似。不是所有的矩阵都可以对角化，但至少在复数域（或任意的代数闭域）内，所有的矩阵都相似于一些被称为若尔当标准形的简单的矩阵。另一种标准形：弗罗贝尼乌斯标准形则在任意的域上都适用。只要查看A和B所对应的标准形是否一致，就能知道两者是否相似。

 

https://en.wikipedia.org/wiki/Matrix_similarity

Motivating example[edit]

When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in ℝ³ when the axis of rotation is not aligned with the coordinate axis can be complicated to compute. If the axis of rotation were aligned with the positive z-axis, then it would simply be

{\displaystyle S={\begin{bmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}},

where {\displaystyle \theta } is the angle of rotation. In the new coordinate system, the transformation would be written as

{\displaystyle y'=Sx'},

where x' and y' are the original and transformed vectors in a new basis containing a vector parallel to the axis of rotation. In the original basis, the transform would be written as

{\displaystyle y=Tx},

where vectors x and y and the unknown transform matrix T are in the original basis. To write T in terms of the simpler matrix, we use the change-of-basis matrix P that transforms x and y as {\displaystyle x'=Px} and {\displaystyle y'=Py}:

{\displaystyle {\begin{aligned}&&y'&=Sx'\\&\Rightarrow &Py&=SPx\\&\Rightarrow &y&=\left(P^{-1}SP\right)x=Tx\end{aligned}}}

Thus, the matrix in the original basis is given by {\displaystyle T=P^{-1}SP}. The transform in the original basis is found to be the product of three easy-to-derive matrices. In effect, the similarity transform operates in three steps: change to a new basis (P), perform the simple transformation (S), and change back to the old basis (P⁻¹).