An effective exact method is proposed for computing generalized eigenspaces of a matrix of integers or rational numbers. Keys of our approach are the use of minimal annihilating polynomials and the concept of the Jourdan-Krylov basis. A new method, called Jordan-Krylov elimination, is introduced to design an algorithm for computing Jordan-Krylov basis. The resulting algorithm outputs generalized eigenspaces as a form of Jordan chains. Notably, in the output, components of generalized eigenvectors are expressed as polynomials in the associated eigenvalue as a variable.

翻译：本文提出了一种有效的精确方法，用于计算整数或有理数矩阵的广义特征空间。该方法的核心在于利用极小湮灭多项式以及Jordan-Krylov基的概念。我们引入了一种称为Jordan-Krylov消元的新方法，以设计计算Jordan-Krylov基的算法。所得算法以Jordan链的形式输出广义特征空间。值得注意的是，在输出中，广义特征向量的分量被表示为以相应特征值为变量的多项式。