Space Of Square Matrices Research Articles

Construction of bases in space M(2,R) in computer mathematics system

In finite-dimensional linear spaces, the question of constructing a basis is related to matrix computations, the labor intensity of which increases not only with increasing dimensionality of spaces, but also with taking into account the nature of this space. In this context, we consider an alternative solution of this question in the space of square matrices of the 2nd order in the system of computer mathematics, which has a very rapid development today.

Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation

We present several algebraic and differential-geometric constructions of tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. In particular, we obtain a family of new (nonlinear) polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang–Baxter equation. Liouville integrability is established for some of these maps. Also, we show how to derive linear tetrahedron maps as linear approximations of nonlinear ones, using Lax representations and the differentials of nonlinear tetrahedron maps on manifolds. We apply this construction to two nonlinear maps: a tetrahedron map obtained in Dimakis and Müller-Hoissen (2019 Lett. Math. Phys. 109 799–827) in a study of soliton solutions of vector Kadomtsev–Petviashvili equations and a tetrahedron map obtained in Konstantinou-Rizos (2020 Nucl. Phys. B 960 115207) in a study of a matrix trifactorisation problem related to a Darboux matrix associated with a Lax operator for the nonlinear Schrödinger equation. We derive parametric families of new linear tetrahedron maps (with nonlinear dependence on parameters), which are linear approximations for these nonlinear ones. Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron map from Sergeev’s classification Sergeev (1998 Lett. Math. Phys. 45 113–9). These matrix generalisations can be regarded as tetrahedron maps in noncommutative variables. Besides, several tetrahedron maps on arbitrary groups are constructed.

An Approach to the Stabilization Problem of a Parametrically Uncertain Linear Nonstationary System

An approach based on the method of predictive models and the method of superstabilization is proposed to solve the problem of stabilization of parametrically uncertain linear nonstationary systems. The parametric uncertainty is specified using a family of compact sets in the space of square matrices. This approach is rigorously justified for second-order systems, but can be generalized to systems of arbitrary order.

Consistent projections and indicators in pairwise comparisons

This study examines several generic properties of weighted consistent projections and indicators of inconsistency in an arbitrary finite dimensional inner product space of square matrices. In the case of weighted Frobenius inner products we present explicit formulae for them in terms of the matrix entries and weights. It extends the recent results, due to Koczkodaj et al. [Fund. Inform. 172 (2020) 379–397], to positive matrices in pairwise comparisons. Moreover, we discuss the possibilities of a proper choice of the inner product weights for numerical applications based on the orthogonal consistent projections.

Многомерная интерполяция

The paper deals with iterative interpolation methods in forms of similar recursive procedures defined by a sort of simple functions (interpolation basis) not necessarily real valued. These basic functions are kind of arbitrary type being defined just by wish and considerations of user. The studied interpolant construction shows virtue of versatility: it may be used in a wide range of vector spaces endowed with scalar product, no dimension restrictions, both in Euclidean and Hilbert spaces. The choice of basic interpolation functions is as wide as possible since it is subdued nonessential restrictions. The interpolation method considered in particular coincides with traditional polynomial interpolation (mimic of Lagrange method in real unidimensional case) or rational, exponential etc. in other cases. The interpolation as iterative process, in fact, is fairly flexible and allows one procedure to change the type of interpolation, depending on the node number in a given set. Linear interpolation basis options (perhaps some nonlinear ones) allow to interpolate in noncommutative spaces, such as spaces of nondegenerate matrices, interpolated data can also be relevant elements of vector spaces over arbitrary numeric field. By way of illustration, the author gives the examples of interpolation on the real plane, in the separable Hilbert space and the space of square matrices with vektorvalued source data.

Matrix Schubert varieties and Gaussian conditional independence models

Matrix Schubert varieties are certain varieties in the affine space of square matrices which are determined by specifying rank conditions on submatrices. We study these varieties for generic matrices, symmetric matrices, and upper triangular matrices in view of two applications to algebraic statistics: We observe that special conditional independence models for Gaussian random variables are intersections of matrix Schubert varieties in the symmetric case. Consequently, we obtain a combinatorial primary decomposition algorithm for some conditional independence ideals. We also characterize the vanishing ideals of Gaussian graphical models for generalized Markov chains. In the course of this investigation, we are led to consider three related stratifications, which come from the Schubert stratification of a flag variety. We provide some combinatorial results, including describing the stratifications using the language of rank arrays and enumerating the strata in each case.

Invariant theory in exterior algebras and Amitsur–Levitzki type theorems

This article discusses invariant theories in some exterior algebras, which are closely related to Amitsur–Levitzki type theorems.First we consider the exterior algebra on the vector space of square matrices of size n, and look at the invariants under conjugations. We see that the algebra of these invariants is isomorphic to the exterior algebra on an n-dimensional vector space. Moreover we give a Cayley–Hamilton type theorem for these invariants (the anticommutative version of the Cayley–Hamilton theorem). This Cayley–Hamilton type theorem can also be regarded as a refinement of the Amitsur–Levitzki theorem.We discuss two more Amitsur–Levitzki type theorems related to invariant theories in exterior algebras. One is a famous Amitsur–Levitzki type theorem due to Kostant and Rowen, and this is related to O(V)-invariants in Λ(Λ2(V)). The other is a new Amitsur–Levitzki type theorem, and this is related to GL(V)-invariants in Λ(Λ2(V)⊕S2(V⁎)).

Characterization of Birkhoff–James orthogonality

The Birkhoff–James orthogonality is a generalization of Hilbert space orthogonality to Banach spaces. We investigate this notion of orthogonality when the Banach space has more structures. We start by doing so for the Banach space of square matrices moving gradually to all bounded operators on any Hilbert space, then to an arbitrary C∗-algebra and finally a Hilbert C∗-module.

Agents Network on a Grid: An Approach with the Set of Circulant Operators

In this work we present some matrix operators named circulant operators and their action on square matrices. This study on square matrices provides new insights into the structure of the space of square matrices. Moreover it can be useful in various fields as in agents networking on Grid or large-scale distributed self-organizing grid systems. Keywords— Pascal matrices, Binomial Recursion, Circulant Operators, Square Matrix Bipartition, Local Network, Parallel networks of agents.

Some Differential-Geometric Remarks on a Method for Minimizing Constrained Functionals of Matrix-Valued Functions

In [3] an approach is given for minimizing certain functionals on certain spaces $${\cal N} = {\rm Maps}(\Omega,N)$$ , where ? is a domain in some Euclidean space and N is a space of square matrices satisfying some extra condition(s), e.g. symmetry and positive-definiteness. The approach has the advantage that in the associated algorithm, the preservation of constraints is built in automatically. One practical use of such an algorithm its its application to diffusion-tensor imaging, which in recent years has been shown to be a very fruitful approach to certain problems in medical imaging. The method in [3] is motivated by differential-geometric considerations, some of which are discussed briefly in [3] and in greater detail in [4]. We describe here certain geometric aspects of this approach that are not readily apparent in [3] or [4]. We also discuss what one can and cannot hope to achieve by this approach.

Formula omitted]-modules associated to the projective space of [formula omitted] matrices

Let us consider X the complex vector space of square matrices and P ( X ) the associated projective space. Denote A the quotient algebra of all SL n ( C ) × SL n ( C ) -invariant differential operators modulo those vanishing on SL n ( C ) × SL n ( C ) -invariant functions. We show that the inverse image functor π + , where π : X \\ { 0 } → P ( X ) is the canonical projection, establishes an equivalence of categories between the category of regular holonomic D -modules on the projective space P ( X ) and the quotient category of graded A -modules of finite type modulo those supported by { 0 } . Then we deduce a combinatorial classification of regular holonomic D P ( X ) -modules. To cite this article: P. Nang, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

On families of square matrices

In this paper we classify families of square matrices up to the following natural equivalence. Thinking of these families as germs of smooth mappings from a manifold to the space of square matrices, we allow arbitrary smooth changes of co-ordinates in the source and pre- and post- multiply our family of matrices by (generally distinct) families of invertible matrices, all dependent on the same variables. We obtain a list of all the corresponding simple mappings (that is, those that do not involve adjacent moduli). This is a non-linear generalisation of the classical notion of linear systems of matrices. We also make a start on an understanding of the associated geometry. 2000 Mathematics Subject Classification 58K40, 58K50, 58K60, 32S25.

A magic decomposition

A lively example to use in a first course in linear algebra to clarify vector space notions is the space of square matrices of fixed order with its subspaces of affine, coaffine, doubly affine, and magic squares. In this note, the projection theorem is illustrated by explicitly constructing the orthogonal projections (in closed forms) of any matrix U onto these subspaces. The results follow directly from a canonical decomposition of U.

Character sums and intersection cohomology complexes associated to the space of square matrices

Let χ be a multiplicative character of a finite field F q . We study by an algebro-geometric method the finite Fourier transforms of χ(det x), where x runs over the square matrices over F q (untwisted case), or the Hermitian matrices (twisted case). We take up these examples as a first step to develop Intersection Cohomology Method in the theory of prehomogeneous vector spaces.

Dykstra's Algorithm for a Constrained Least-squares Matrix Problem

We apply Dykstra's alternating projection algorithm to the constrained least-squares matrix problem that arises naturally in statistics and mathematical economics. In particular, we are concerned with the problem of finding the closest symmetric positive definite bounded and patterned matrix, in the Frobenius norm, to a given matrix. In this work, we state the problem as the minimization of a convex function over the intersection of a finite collection of closed and convex sets in the vector space of square matrices. We present iterative schemes that exploit the geometry of the problem, and for which we establish convergence to the unique solution. Finally, we present preliminary numberical results to illustrate the performance of the proposed iterative methods.

Spatial decomposition of functionally commutative matrices

Let I ⊆ R and D ⊆ C , where R and C are the fields of real and complex numbers, respectively. Let C n×n be the space of square matrices of order n over C . A matrix-valued function F : I → C n×n is said to be proper on I if F ( t ) = f ( t , A ), where A ϵ C n×n and f : I × D → C is a scalar function, and F is said to be semiproper on I if F ( t ) F ( t ) = F ( τ ) F ( t ) for all t , τ ϵ I . The main results presented here are: (1) a characterization together with some potentially useful results for proper matrix functions; (2) a characterization of semiproper matrix functions in terms of proper ones; (3) a new and systematic procedure for decomposing a semiproper matrix function into a finite sum of mutually commutative proper ones. Some important applications of these new results in control engineering, linear systems theory, and the theory of linear differential equations are also included.