“矩阵相似”等价于“特征矩阵等价”的证明

两个数域上的数字矩阵的相似问题可以转化为其相应的特征矩阵等价的命题来解决。很多教科书对这一问题的证明过于简单,没有真正的区分数字矩阵和多项式矩阵之间的不同。数字矩阵与多项式矩阵的区别就在于数字矩阵经过加法、减法、乘法、除法后还是数字矩阵,但多项式矩阵不能无条件的进行除法运算后还是多项式矩阵。所以,我们在证明多项式矩阵的有些问题时,不能直接套用数字矩阵的一些命题和定理。本文对"数字矩阵相似"等价于"特征矩阵等价"这一问题进行了详细论述。

The Proof of“Matrix’ s Similarity” Equivalent to“The Equivalence of Characteristic Matrix”

The problem about digital matrix’s similarity can be solved with the equivalence of the corresponding characteristic matrix.In many textbooks the proof of this problem is too simple; they haven’ t made a distinction between the digital matrix and the polynomial matrix.The difference between digital matrix and polynomial matrix is that the digital matrix is still digital matrix through the operation of addition, subtraction, multiplication, and division, but the polynomial matrix is not.So, when we prove some problems about polynomial matrix, it can not be used directly with the digital matrix’ s definitions and theorems.This article has discussed in detail this problem that "digital matrix’ s similarity” is equivalent to “the equivalence of characteristic matrix”.