Jordan Normal and Rational Normal Form Algorithms

TL;DR

This paper introduces a deterministic algorithm for computing the Jordan normal form based on the Fadeev formula, which differs from traditional methods by not relying on Frobenius or Smith normal forms, and has been implemented in computer algebra systems.

Contribution

The paper presents a novel rational Jordan normal form algorithm using the Fadeev formula, offering a different approach from traditional algorithms based on Frobenius or Smith forms.

Findings

Algorithm complexity is O(n^4) with known characteristic polynomial factorization.

Implementation available in Maple and Giac/Xcas systems.

Algorithm effectively computes Jordan normal form using eigenvector properties from the Fadeev formula.

Abstract

In this paper, we present a determinist Jordan normal form algorithms based on the Fadeev formula: \[(\lambda \cdot I-A) \cdot B(\lambda)=P(\lambda) \cdot I\] where $B(\lambda)$ is $(\lambda \cdot I-A)$'s comatrix and $P(\lambda)$ 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 $B(\lambda_0)$ are eigenvectors of $A$ associated to $\lambda_0$ for any root $\lambda_0$ 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.