🤖 AI Summary
This paper addresses the exact computation of generalized eigenspaces for integer/rational matrices. We propose a symbolic algorithm based on minimal annihilating polynomials and Jordan–Krylov bases. Methodologically, we introduce Jordan–Krylov elimination—the first technique to explicitly express each component of a generalized eigenvector as a polynomial function of the corresponding eigenvalue—and perform all computations symbolically using exact linear algebra, thereby avoiding rounding errors entirely. Our contributions are threefold: (1) the first exact algorithm producing structurally explicit Jordan chains; (2) an algebraic, closed-form expression for generalized eigenvectors; and (3) support for symbolic-level analysis and subsequent algebraic reasoning. Experimental evaluation confirms the method’s effectiveness and reliability on medium- and small-scale rational matrices.
📝 Abstract
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.