Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases

Abstract

In this paper we present two algorithms for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gröbner bases. This can be viewed as the preprocessing for the computation of Jacobson normal form and also used for the computation of Smith normal form in the commutative case. We propose a general framework for handling, among other, operator algebras with rational coefficients. We employ special "polynomial" strategy in Ore localizations of non-commutative G-algebras and show its merits. In particular, for a given matrix M we provide an algorithm to compute U, V and D with fraction-free entries such that U M V = D holds. The polynomial approach allows one to obtain more precise information, than the rational one e. g. about singularities of the system.Our implementation of polynomial strategy shows very impressive performance, compared with methods, which directly use fractions. In particular, we experience quite moderate swell of coefficients and obtain uncomplicated transformation matrices. This shows that this method is well suitable for solving nontrivial practical problems. We present an implementation of algorithms in Singular:Plural and compare it with other available systems. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a bigger class of non-commutative algebras.

Algebraic Controllability of Nonlinear Mechanical Control Systems

“…is called left prime if there exist matrices Note that there are some previous studies of algorithms for the computation of the Jacobson form of a polynomial matrix over a non-commutative Euclidean domain [3]- [6]. For the computation of the Jacobson form, we can apply the library called "Jacobson.lib" of the computer algebra system SINGU-LAR::PLURAL.…”

“…If the differential operator ring is a non-commutative Euclidean domain, hyperregularity of the polynomial matrix can be examined by repeating elementary matrix operations. Furthermore, computer algebra systems [3]- [6] can be applied to study hyper-regularity of the polynomial matrix. Moreover, algebraic controllability of a mechanical control system which can be transformed into affine first order differential equations relates to strong accessibility, which characterized by the Lie rank condition, defined in [7].…”

Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gröbner bases

“…As in Levandovskyy and Schindelar (2011), we continue working with Ore localizations of G-algebras, which are principal ideal domains. A G-algebra R is a Noetherian integral domain, hence there exists its total two-sided ring of fractions Quot(R) = (R \ {0}) −1 R, which is a division ring (skew field).…”

“…As for examples, a 2 × 2 matrix over the Weyl algebra has been considered in detail in Example 3.8 of (Levandovskyy and Schindelar, 2011). Note that a fraction-free method was used indeed.…”

“…In (Levandovskyy and Schindelar, 2011), we compared our implementation with the one by D. Robertz. In turned out that with our implementation, one experiences moderate swell of coefficients and obtains uncomplicated transformation matrices.…”