模观点下看矩阵Jordan标准形的存在性

Study the Existence of the Jordan Canonical Form of a Complex Matrix from the Module Viewpoint

利用模的有关性质,把复数域上n维向量空间V看成是一个V上线性变换T所决定的C[x]—模,将有限加群的结构定理推广到C[x]—模V中,得到了C[x]—模V可以分解成一些循环C[x]—模的直和,进而得到任一个复矩阵都与一Jordan标准形相似。这样,在模观点下矩阵相似于若当形矩阵就与C[x]—模V分解成一些循环C[x]—子模的直和是相当的。

Using the properties of modules and regarding the n-dimensional vector space V as a C-module determined by a linear transformation T in V , the author extends the theorem of the structure of the finite additive group to the C-module, and proves that the C-module V is a direct sum of cyclic modules, furthermore, obtains that any complex matrix is similar to a Jordan canonical form.