关于整数矩阵几个问题的思考

The Research on the Problems of Integer Matrix

由于环中的元素未必有逆,因此域上的矩阵的那些结果在交换环上的矩阵就未必能够成立。在前人研究整数矩阵可逆的等价条件、整数矩阵的初等变换、整系数线性方程组解的判定、整数矩阵的应用的基础上,进一步提出整数矩阵的特征值、特征向量、相似以及相似对角化等问题,并得出了一系列结果。主要结果有整数矩阵仅通过整初等行变换一定可变成上三角矩阵(下三角矩阵);整特征向量对应的特征值一定是整数;对称整数矩阵的特征值与特征向量的关系;整数矩阵与对角矩阵整相似的两个充要条件;对称整数矩阵A有n个不同的特征值,且A可对角化,则A一定是整对角矩阵。

Because the element on the metal ring may not have the inverse, so the theorems on the matrix in field would perhaps incorrect. In this article, I mainly do research on the eigenvalue, eigenvector and similar diagonalization. These results are.. integer matrix can change into upper triangular matrix through integer elementary row operation; two necessary and sufficient conditions about similarity diagonalization of integer matrix; if the integer matrix has different eigenvalue, it can diagonalizable if and only if it is an integer diagonal matrix.