Kronecker Invariants Research Articles

Bounded rank perturbations of a regular matrix pencil

In this paper we study the possible Kronecker invariants of an arbitrary matrix pencil obtained by bounded rank perturbation of a regular matrix pencil. We solve this problem in the case of a perturbation of minimal rank.

Jordan–Kronecker Invariants for Lie Algebras of Small Dimensions

In this paper, Jordan–Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space 𝔤 proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A − λB)(x), we consider Jordan–Kronecker invariants for this pencil and how they change with x (the latter is done for 6-dimensional Lie algebras).

Matrix pencils completions under double rank restrictions

In this paper we study the General Matrix Pencil Completion Problem under double rank restrictions. Considered rank restrictions are not structural, hence the obtained results deal with full sets of Kronecker invariants of the involved matrix pencils, and they generalize many of the existing results in the literature, for example [1, 5, 7, 9, 11, 12, 21, 24, 45, 46]. Main methods consist of combining the celebrated S?-Thompson?s result [1, 39, 44] with novel results on rank restrictions in completions of matrix pencils. All of the obtained results are explicit and constructive.

Jordan–Kronecker Invariants of Semidirect Sums of the Form sl(n) + (ℝn)k and gl(n) + (ℝn)k

We calculate Jordan–Kronecker invariants for semidirect sums of Lie algebras sl(n) and gl(n) with k copies of ℝn with respect to their standard representation for cases where k > n or n is a multiple of k.

Completion of matrix pencils with a single rank restriction

In this paper we consider and solve the General Matrix Pencil Completion Problem (GMPCP) under single rank restrictions. These constraints are less limiting than any other restrictions on the GMPCP studied in the past. Thus, the obtained results directly generalize the existing results in the literature on the GMPCP. Finally, as one direct corollary of the main results, we obtain a novel result on determining the possible Kronecker invariants of a quasi-regular matrix pencil with a prescribed subpencil.

JORDAN–KRONECKER INVARIANTS OF LIE ALGEBRA REPRESENTATIONS AND DEGREES OF INVARIANT POLYNOMIALS

For an arbitrary representation ρ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan–Kronecker invariants of ρ. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of ρ. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.

Orientation of Implicit State Space Models and the Partitioning of Kronecker Structure

Early stages modelling of processes involves issues of classification of variables into inputs, outputs and internal variables, referred to as Model Orientation Problem (MOP) which may be addressed on state space implicit, or matrix pencil descriptions. Defining orientation is equivalent to producing state space models of the regular or singular type. In this paper we consider autonomous differential descriptions defined by matrix pencils and then search for strict equivalence transformations which introduce the partitioning of the implicit vector into states and possible inputs and outputs, referred to as system orientation. The Kronecker invariant structure of the matrix pencil description is shown to be central to the solution of system orientation and this is expressed as a problem of classification and partitioning of the Kronecker invariants. It is shown that the types of Kronecker invariants characterise the nature of the system orientation solutions. Studying the conditions, under which such oriented models may be derived, as well as their structural properties in terms of the Kronecker structure, is the issue considered here.

Jordan-Kronecker invariants of Borel subalgebras of semisimple Lie algebras

In the theory of bi-Hamiltonian systems, the generalized Mischenko–Fomenko conjecture is known. The conjecture states that there exists a complete set of polynomial functions in involution with respect to a pair of naturally defined Poisson structures on a dual space of a Lie algebra. This conjecture is closely related to the argument shift method proposed by A. S. Mishchenko and A. T. Fomenko in [10]. In research works devoted to this conjecture, a connection was found between the existence of a complete set in bi-involution and the algebraic type of the pencil of compatible Poisson brackets, defined by a linear and constant bracket. The numbers that describe the algebraic type of the generic pencil of brackets on the dual space to a Lie algebra are called the Jordan–Kronecker invariants of a Lie algebra. The notion of Jordan–Kronecker invariants was introduced by A. V. Bolsinov and P. Zhang in [2]. For some classes of Lie algebras (for example, semisimple Lie algebras and Lie algebras of low dimension), the Jordan–Kronecker invariants have been computed, but in the general case the problem of computation of the Jordan–Kronecker invariants for an arbitrary Lie algebra remains open. The problem of computation of the Jordan–Kronecker invariants is frequently mentioned among the most interesting unsolved problems in the theory of integrable systems [4, 5, 6, 11]. In this paper, we compute the Jordan–Kronecker invariants for the series 𝐵𝑠𝑝(2𝑛) and construct complete sets of polynomials in bi-involution for each algebra of the series. Also, we calculate the Jordan–Kronecker invariants for the Borel subalgebras 𝐵𝑠𝑜(𝑛) for any 𝑛. Thus, together with the results obtained in [2] for 𝐵𝑠𝑙(𝑛), this paper presents a solution to the problem of computation of Jordan–Kronecker invariants for Borel subalgebras of classical Lie algebras.

GMPCP for one row and one column completion

In this paper, the possible Kronecker invariants of a matrix pencil obtained by simultaneous one row and one column completion of a given matrix pencil are characterized. This presents a new approach to the General Matrix Pencil Completion Problem (GMPCP), where simultaneous row and column completion is considered.

Complete characterisation of Kronecker invariants of a matrix pencil with a prescribed quasi-regular subpencil


 
 
 In this paper, the possible Kronecker invariants of a matrix pencil with a prescribed quasi-regular subpencil are determined. 
 
 

Rank One Perturbations of Matrix Pencils

For the first time in the literature we completely resolve the open problem of describing the possible Kronecker invariants of an arbitrary matrix pencil under rank one perturbations. The solution ...

Model specification and selection for multivariate time series

Three major difficulties are identified with an established echelon form approach (see Hannan (1987)) to specifying a Vector Autoregressive Moving Average, VARMA, model for an observed time series. A family of state space representations, valid for each integer, h, is introduced, and collectively referred to as multistep state space representations. This family includes as its special case, with h=0, a state space representation introduced earlier by Akaike (1974), and, with h=1, that introduced by Cooper and Wood (1982). Appropriate generalizations of the notions of minimality, McMillan degree, left matrix fraction description and Kronecker indices, as applicable individually to each member of this family, are presented. The reverse echelon form and state space representation corresponding to the Kronecker indices for each h are derived, and the former illustrated with three examples of standard VARMA processes. The question of how the presence of zero constraints on the coefficients of a reverse echelon form may be detected solely from an inspection of the Kronecker indices is examined. A canonical correlation procedure proposed originally by Akaike (1976) for h=0 is considered for estimating the Kronecker indices with each h. The efficacy of the estimation procedure is investigated by a simulation study. A procedure is suggested for implementing the new approach introduced in this paper with an observed time series, and three different applications of this approach are outlined. This approach is also related to some of its alternatives, including the Kronecker invariants of Poskitt (1992) and the scalar component approach of Tiao and Tsay (1989).

The General Matrix Pencil Completion Problem: A Minimal Case

In this paper we resolve a classical, long-standing open general matrix pencil completion problem in a minimal case. The problem consists of describing the possible Kronecker invariants of a pencil...

Submodules of Kronecker modules via extension monoid products

Using some results on Reineke's extension monoid product over the Kronecker algebra kQ, we provide numerical criteria in terms of Kronecker invariants for a module in mod-kQ to be isomorphic with the submodule of an another module in mod-kQ. The results can be transferred to matrix pencils in solving an important challenge in matrix pencil theory: characterize in terms of Kronecker invariants when a given pencil is a subpencil of an another one.

Jordan–Kronecker invariants for semidirect sums defined by standard representation of orthogonal or symplectic Lie algebras

We calculate Jordan–Kronecker invariants for semi-direct sums of Lie algebras so(n) and sp(n) with several copies of R n with respect to their standard representation.

A short solution to the subpencil problem involving only column minimal indices

We give an explicit numerical characterization in terms of classical Kronecker invariants of the subpencil relation between two matrix pencils determined only by minimal indices for columns (respectively for rows). In addition, a method to construct the completion pencils is also presented (thus solving the pencil completion problem completely in this special case). We showcase our method by carrying out the computations in a well-explained example. Theoretically, our criteria could also be obtained as a particular case of the very technical and involved results by Dodig and Stošić (from [6] and [5]). However, our self-contained approach is easy to follow and independent of the technical results used in the mentioned papers.

Finite-dimensional integrable systems: A collection of research problems

This article suggests a series of problems related to various algebraic and geometric aspects of integrability. They reflect some recent developments in the theory of finite-dimensional integrable systems such as bi-Poisson linear algebra, Jordan–Kronecker invariants of finite dimensional Lie algebras, the interplay between singularities of Lagrangian fibrations and compatible Poisson brackets, and new techniques in projective geometry.

Completion of quasi-regular matrix pencils

In this paper we give solutions for two important particular cases of the general completion problem, involving quasi-regular matrix pencil. We describe the possible Kronecker invariants of a quasi-regular pencil whose subpencil is prescribed in the case when the prescribed subpencil is without nontrivial homogeneous invariant factors. Also, we solve “the dual problem” by describing the possible Kronecker invariants of a pencil without nontrivial homogeneous invariant factors when a quasi-regular subpencil is prescribed. These two results are expected to be the crucial steps toward a solution of the general completion problem. The results are given over algebraically closed fields.

Vector autoregressive moving average identification for macroeconomic modeling: A new methodology

This paper develops a new methodology for identifying the structure of VARMA time series models. The analysis proceeds by examining the echelon form and presents a fully automatic, strongly consistent, data driven approach to model specification. A novel feature of the inferential procedures developed here is that they work in terms of a canonical representation based on the Kronecker invariants in which the variables are expressed in the form of scalar dynamic structural equations derived from the VARMA system. This feature facilitates the construction of procedures which, from the perspective of macroeconomic modeling, can be efficacious in that they do not rely on VAR approximations. Techniques that are applicable to both asymptotically stationary and unit-root, partially nonstationary (cointegrated) time series models are presented. The inferential potential of the techniques is illustrated via simulation experiments that use data generating mechanisms based on real world examples drawn from the time series literature. Aspects of the Kronecker invariants that impinge on practical application and that have not hitherto been discussed in the literature are explored.

JORDAN–KRONECKER INVARIANTS OF FINITE-DIMENSIONAL LIE ALGEBRAS

For any finite-dimensional Lie algebra we introduce the notion of Jordan–Kronecker invariants, study their properties, and discuss examples. These invariants naturally appear in the framework of the bi-Hamiltonian approach to integrable systems on Lie algebras and are closely related to Mischenko–Fomenko’s argument shift method. We also state a generalised argument shift conjecture and prove it for many series of Lie algebras.