特征根全部为整数的整矩阵的判别及其应用

The Discrimination and Application of Integral Matrix with all Integral Eigenvalues

利用西尔维斯特定理以及整矩阵的上三角化引理证明了一个整矩阵的特征根全部为整数的充要条件是该整矩阵可表示为若干个特殊整矩阵的和.应用这个结论可以构造有特定特征值的整矩阵以及判断一个矩阵是否与整矩阵相似.

We use sylvester theorem and upper-triangulation lemma to prove that an integral matrix has all integral eigenvalues if and only if the integral matrix can be expressed as a sum of some special integral matrices.With the method,we can construct an integral matrix with specific eigenvalues and judge whether a matrix is similar to an integral matrix.