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Jordan Normal and Rational Normal Form Algorithms

Bernard Parisse·Morgane Vaughan
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Abstract

In this paper, we present a determinist Jordan normal form algorithms based on the Fadeev formula: \[(λ\cdot I-A) \cdot B(λ)=P(λ) \cdot I\] where $B(λ)$ is $(λ\cdot I-A)$'s comatrix and $P(λ)$ is $A$'s characteristic polynomial. This rational Jordan normal form algorithm differs from usual algorithms since it is not based on the Frobenius/Smith normal form but rather on the idea already remarked in Gantmacher that the non-zero column vectors of $B(λ_0)$ are eigenvectors of $A$ associated to $λ_0$ for any root $λ_0$ of the characteristical polynomial. The complexity of the algorithm is $O(n^4)$ field operations if we know the factorization of the characteristic polynomial (or $O(n^5 \ln(n))$ operations for a matrix of integers of fixed size). This algorithm has been implemented using the Maple and Giac/Xcas computer algebra systems.

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Bernard Parisse, Morgane Vaughan. 2004-12-02. Jordan Normal and Rational Normal Form Algorithms. https://arxiv.org/abs/cs/0412005

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