[Paper Review] Jordan Normal and Rational Normal Form Algorithms

Bernard Parisse, Morgane Vaughan|arXiv (Cornell University)|Dec 2, 2004

Polynomial and algebraic computation7 references3 citations

TL;DR

This paper presents a deterministic algorithm for computing the Jordan normal form and rational normal form of a matrix using the Fadeev formula and the comatrix of (λI − A). It leverages the fact that non-zero columns of B(λ₀) are eigenvectors for eigenvalue λ₀, enabling efficient computation via polynomial matrix operations with O(n⁴) field operations when the characteristic polynomial is factored.

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.

Motivation & Objective

• To develop a deterministic algorithm for computing the Jordan normal form of a matrix over an algebraically closed field.
• To extend the method to compute the rational normal form over arbitrary fields without field extensions.
• To provide an alternative to Frobenius/Smith normal form-based approaches by using the comatrix of (λI − A).
• To achieve O(n⁴) complexity under known characteristic polynomial factorization, making it suitable for parallelization.
• To implement and integrate the algorithm into computer algebra systems like Maple and Giac/Xcas for practical use.

Proposed method

• Utilizes the Fadeev algorithm to compute the characteristic polynomial P(λ) and the matrix polynomial B(λ), the comatrix of (λI − A).
• Employs the identity (λI − A)B(λ) = P(λ)I to derive recurrence relations for B(λ) coefficients using trace-based formulas.
• Applies Hörner's method for efficient evaluation of B(λ₀) at eigenvalues λ₀ to extract eigenvectors.
• Constructs generalized eigenvectors by solving linear systems involving Q(A), where Q is an irreducible factor of P(λ).
• Uses binomial coefficient identities to reconstruct higher-order cyclic vectors from base vectors via recurrence (9).
• Implements a hybrid method in Maple and Giac/Xcas that combines complex Jordan form computation with conversion to rational normal form for irreducible factors of degree >1.

Experimental results

Research questions

• RQ1 Can the Jordan normal form be computed deterministically using only the comatrix of (λI − A) and eigenvalue evaluation?
• RQ2 How can the rational normal form be constructed without field extensions, using only matrix polynomial operations?
• RQ3 What is the computational complexity of a comatrix-based algorithm compared to Frobenius/Smith normal form methods?
• RQ4 Can the algorithm be efficiently parallelized or implemented in computer algebra systems with minimal symbolic overhead?
• RQ5 How can generalized eigenvectors be systematically reconstructed from base eigenvectors using polynomial matrix operations?

Key findings

• The algorithm computes the Jordan normal form in O(n⁴) field operations when the characteristic polynomial is fully factored.
• For integer matrices of fixed size, the complexity increases to O(n⁵ ln n) due to polynomial factorization costs.
• The rational normal form is constructed by first computing a pseudo-rational form and then converting it via a structured transformation.
• The method avoids reliance on Frobenius or Smith normal forms, offering a new pathway to rational canonical form computation.
• The algorithm has been successfully implemented in Maple and natively in Giac/Xcas, with functions like jordan and rat_jordan available for direct use.
• A key insight is that non-zero columns of B(λ₀) are eigenvectors for eigenvalue λ₀, enabling direct eigenvector extraction without solving (A − λ₀I)v = 0.