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：提出了一种有效精确方法，用于计算整数或有理数矩阵的广义特征空间。该方法的核心在于使用极小消零多项式与约当-克雷洛夫基概念。通过引入新方法——约当-克雷洛夫消元法，设计出计算约当-克雷洛夫基的算法。最终算法以约当链形式输出广义特征空间。值得关注的是，输出结果中广义特征向量的分量被表示为关于相关特征值变量的多项式。