Jordan Normal and Rational Normal Form Algorithms

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

Jordan Normal and Rational Normal Form Algorithms

In this paper, we present a determinist Jordan normal form algorithms based on the Fadeev formula: (λ · I − A) · B(λ) = P (λ) · I where B(λ) is (λ · 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(λ0) are eigenvectors of A associated to λ0 for any root λ0 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.