Search

In the present paper we shall obtain the following result: Let n ≥ 2 and m ≥ 1 be integers. Let p (x 1 , ... , x m) be a polynomial with zero constant term in non-commutative variables over an algebraically closed field K. Suppose that 1 ≤ r ≤ n − 1 , where r is the order of p. Then p (T n (K)) is a dense subset of T n (K) (r − 1) (with respect to the Zariski topology). This is a solution of a problem presented by Gargate and de Mello and a supplement to a result obtained by Panja and Prasad in 2023. [ABSTRACT FROM AUTHOR]

Galois duals of Multi-twisted (MT) codes are considered in this study. We describe a MT code C as a module over a principal ideal domain. Hence, C has a generator polynomial matrix (GPM) G that satisfies an identical equation. We prove a GPM formula for the Euclidean dual C ⊥ using the identical equation of the Hermite normal form of G . Next, we aim to replace the Euclidean dual with the Galois dual. The Galois inner product is an asymmetric form, so we distinguish between the right and left Galois duals. We show that the right and left Galois duals of a MT code are MT as well but with possibly different shift constants. Some interconnected identities for the right and left Galois duals of a linear code are established and we also introduce the two-sided Galois dual. We use a condition that makes the two-sided Galois dual of a MT code MT, then we describe its GPM. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. A necessary and sufficient condition is established for the equality of the right and left Galois duals of any linear code. [ABSTRACT FROM AUTHOR]

The systems of nonlinear differential equations of certain types can be simplified to matrix forms. Two types of matrix differential equations will be considered in the paper, one is Fermat type matrix differential equation where n = 2 and n = 3, another is Malmquist type matrix differential equation , where α (≠ 0), β, γ are constants. By solving the systems of nonlinear differential equations, we obtain some properties on the meromorphic matrix solutions of the above matrix differential equations. In addition, we also consider two types of nonlinear differential equations, one of them is called Bi-Fermat differential equation. [ABSTRACT FROM AUTHOR]

In this paper we study the backward stability of running a backward stable eigenstructure solver on a pencil S (λ) that is a strong linearization of a rational matrix R (λ) expressed in the form R (λ) = D (λ) + C (λ I ℓ - A) - 1 B , where D (λ) is a polynomial matrix and C (λ I ℓ - A) - 1 B is a minimal state-space realization. We consider the family of block Kronecker linearizations of R (λ) , which have the following structure S (λ) : = M (λ) K ^ 2 T C K 2 T (λ) B K ^ 1 A - λ I ℓ 0 K 1 (λ) 0 0 , where the blocks have some specific structures. Backward stable eigenstructure solvers, such as the QZ or the staircase algorithms, applied to S (λ) will compute the exact eigenstructure of a perturbed pencil S ^ (λ) : = S (λ) + Δ S (λ) and the special structure of S (λ) will be lost, including the zero blocks below the anti-diagonal. In order to link this perturbed pencil with a nearby rational matrix, we construct in this paper a strictly equivalent pencil S ~ (λ) = (I - X) S ^ (λ) (I - Y) that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix R ~ (λ) = D ~ (λ) + C ~ (λ I ℓ - A ~) - 1 B ~ , where D ~ (λ) is a polynomial matrix with the same degree as D (λ) . Moreover, we bound appropriate norms of D ~ (λ) - D (λ) , C ~ - C , A ~ - A and B ~ - B in terms of an appropriate norm of Δ S (λ) . These bounds may be, in general, inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny, by making the matrices appearing in both S (λ) and R (λ) have norms bounded by 1. Thus, for this scaled representation, we prove that the staircase and the QZ algorithms compute the exact eigenstructure of a rational matrix R ~ (λ) that can be expressed in exactly the same form as R (λ) with the parameters defining the representation very near to those of R (λ) . This shows that this approach is backward stable in a structured sense. Several numerical experiments confirm the obtained backward stability results. [ABSTRACT FROM AUTHOR]

In this paper we present a class of convergent infinite products ∏ k = 1 ∞ H k = lim k → ∞ (H k H k − 1 ⋯ H 1) of 3 × 3 hollow matrices of the form H k := [ 0 a k a k b k 0 b k c k c k 0 ] , (0 < a k , b k , c k < 1 / 2). These sequences of hollow matrices are obtained from the Ando-Li-Mathias matrix geometric mean of 2 × 2 positive definite matrices. We further provide a three dimensional analytic manifold and an analytic self-map realizing such sequences. [ABSTRACT FROM AUTHOR]

We introduce a new family of linearizations of rational matrices, which we call block full rank linearizations. The theory of block full rank linearizations is useful as it establishes very simple criteria to determine if a pencil is a linearization of a rational matrix in a target set or in the whole underlying field, by using rank conditions. Block full rank linearizations allow us to recover locally information about zeros and poles. To recover the pole-zero information at infinity, we will define the grade of the new block full rank linearizations as linearizations at infinity and the notion of degree of a rational matrix will be used. Moreover, the eigenvectors of a rational matrix associated with its eigenvalues in a target set can be obtained from the eigenvectors of its block full rank linearizations in that set. This new family of linearizations generalizes and includes the structures appearing in most of the linearizations for rational matrices constructed in the literature. As example, we study the structure and properties of the linearizations in [P. Lietaert et al., Automatic rational approximation and linearization of nonlinear eigenvalue problems, 2021]. [ABSTRACT FROM AUTHOR]

Let A 1 , ... , A s be unitary commutative rings which do not have non-trivial idempotents and let A = A 1 ⊕ ⋯ ⊕ A s be their direct sum. We describe all idempotents in the 2 × 2 matrix ring M 2 (A [ [ X ] ]) over the ring A [ [ X ] ] of formal power series with coefficients in A and in an arbitrary set of variables X. We apply this result to the matrix ring M 2 ( Z n [ [ X ] ]) over the ring Z n [ [ X ] ] where Z n ≅ Z / n Z for an arbitrary positive integer n greater than 1. Our proof is elementary and uses only the Cayley-Hamilton theorem (for 2 × 2 matrices only) and, in the special case A = Z n , the Chinese remainder theorem and the Euler-Fermat theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to initiate the study of the images of non-multilinear polynomials on upper triangular matrix algebras. We shall give a complete description of the images of completely homogenous polynomials on 2 × 2 upper triangular matrix algebras. As a consequence, we show that the image of some special completely homogenous polynomials on 2 × 2 upper triangular matrix algebras are not vector spaces. [ABSTRACT FROM AUTHOR]

In this paper we establish some Positivstellensätze for polynomial matrices, applying the Scherer–Hol theorem. Firstly, we give a representation for polynomial matrices positive definite on subsets of compact polyhedra. Then we establish a Putinar-Vasilescu Positivstellensatz for polynomial matrices. Next we propose a matrix version of the Dickinson–Povh Positivstellensatz. Finally, we establish a version of Marshall's theorem for polynomial matrices, approximating positive semi-definite polynomial matrices using sums of squares. [ABSTRACT FROM AUTHOR]

Let D be the unit disk. Kutzschebauch and Studer (Bull. Lond. Math. Soc. 51 (2019) 995–1004) recently proved that, for each continuous map A:D¯→SL(2,C), which is holomorphic in D, there exist continuous maps E,F:D¯→sl(2,C), which are holomorphic in D, such that A=eEeF. Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible. [ABSTRACT FROM AUTHOR]

We show that any element of the special linear group SL2(R) is a product of two exponentials if the ring R is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to SL2(C) is not surjective. Our result extends to the linear group GL2(R), where it is sharp as well. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe the images of non-commutative multilinear polynomials on upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

Let R be a commutative unital ring. A well-known factorization problem is whether any matrix in SL n (R) is a product of elementary matrices with entries in R. To solve the problem, we use two approaches based on the notion of the Bass stable rank and on construction of a null-homotopy. Special attention is given to the case, where R is a ring or Banach algebra of holomorphic functions. Also, we consider a related problem on representation of a matrix in GL n (R) as a product of exponentials. [ABSTRACT FROM AUTHOR]

In the paper we study probability measure on real-orthogonal projections acting on complex Euclidean space. We proof an analog of Gleason's theorem. We study Hermitian measure and some special class of measures on real-orthogonal projections. [ABSTRACT FROM AUTHOR]

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non‐commuting variables. In this paper we introduce the class of nc‐manifolds, the mathematical objects that at each point possess a neighborhood that has the structure of an nc‐domain in the d‐dimensional nc‐universeMd. We illustrate the use of such manifolds in free analysis through the construction of the non‐commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non‐commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in M2 we construct a two‐dimensional non‐commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton–Girard formulae for power sums of two non‐commuting variables. [ABSTRACT FROM AUTHOR]

We show a diagonalisation variant of Lindström calculation method. As an application of this result we calculate the dimension of the affine space of Negami's splitting matrices. We do so by writing down an expression for the Möbius function in term of Möbius algebra identities. As a corollary we get Lindström's result in a self contained way. Finally, we calculate the partition lattice characteristic polynomial via the Negami's polynomial. [ABSTRACT FROM AUTHOR]

We introduce a new family of strong linearizations of matrix polynomials—which we call “block Kronecker pencils”—and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to a block Kronecker pencil, such as the staircase algorithm for singular pencils or the QZ algorithm for regular pencils. This stability analysis allows us to identify those block Kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial. The global backward error analysis in this work presents for the first time the following key properties: it is a rigorous analysis valid for finite perturbations (i.e., it is not a first order analysis), it provides precise bounds, it is valid simultaneously for a large class of linearizations, and it establishes a framework that may be generalized to other classes of linearizations. These features are related to the fact that block Kronecker pencils are a particular case of the new family of “strong block minimal bases pencils”, which are robust under certain perturbations and, so, include certain perturbations of block Kronecker pencils. [ABSTRACT FROM AUTHOR]

The standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to embed the matrix coefficients of the polynomial into a matrix pencil, transforming the problem into an equivalent generalized eigenvalue problem. Such pencils are known as linearizations. Many of the families of linearizations for matrix polynomials available in the literature are extensions of the so-called family of Fiedler pencils. These families are known as generalized Fiedler pencils, Fiedler pencils with repetition, and generalized Fiedler pencils with repetition—or Fiedler-like pencils for simplicity. The goal of this work is to unify the Fiedler-like pencils approach with the more recent one based on strong block minimal bases pencils introduced in F.M. Dopico et al. (2017) [17] . To this end, we introduce a family of pencils that we have named extended block Kronecker pencils, whose members are, under some generic nonsingularity conditions, strong block minimal bases pencils, and show that, with the exception of the non-proper generalized Fiedler pencils, all Fiedler-like pencils belong to this family modulo permutations. As a consequence of this result, we obtain a much simpler theory for Fiedler-like pencils than the one available so far. Moreover, we expect this simplification to allow for further developments in the theory of Fiedler-like pencils such as global or local backward error analyses and eigenvalue conditioning analyses of polynomial eigenvalue problems solved via Fiedler-like linearizations. [ABSTRACT FROM AUTHOR]

In this paper, given a field with an involutory automorphism, we introduce the notion of Moore–Penrose field by requiring that all matrices over the field have Moore–Penrose inverse. We prove that only characteristic zero fields can be Moore–Penrose, and that the field of rational functions over a Moore–Penrose field is also Moore–Penrose. In addition, for a matrix with rational functions entries with coefficients in a field K , we find sufficient conditions for the elements in K to ensure that the specialization of the Moore–Penrose inverse is the Moore–Penrose inverse of the specialization of the matrix. As a consequence, we provide a symbolic algorithm that, given a matrix whose entries are rational expression over C of finitely many meromorphic functions being invariant by the involutory automorphism, computes its Moore–Penrose inverve by replacing the functions by new variables, and hence reducing the problem to the case of matrices with complex rational function entries. [ABSTRACT FROM AUTHOR]

Livernet and Loday constructed a polarization of the nonsymmetric associative operadwith one operation into a symmetric operadwith two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of, which includes Poisson algebras. We combine this with the dendriform splitting of an associative operation into the sum of two nonassociative operations, and use Koszul duality for quadratic operads, to construct one-parameter deformations of the nonsymmetric dendriform and diassociative operads into the category of symmetric operads. [ABSTRACT FROM PUBLISHER]

A criterion is presented that characterizes when two matrix polynomials of any size, rank and degree have the same finite and infinite elementary divisors. This characterization inherits a coprimeness condition of the extended unimodular equivalence defined by Pugh and Shelton [17] in the set of real or complex matrix polynomials satisfying the constraint that the difference between the number of rows and columns is constant. This extended unimodular equivalence is first generalized to matrices of any size with elements in any principal ideal domain. [ABSTRACT FROM AUTHOR]

The concept of coprimeness of matrices with elements in a field of fractions is introduced. We focus on the field of rational functions and define when two rational matrices are coprime with respect to different rings. The definition of coprimeness at infinity of polynomial matrices is obtained as a particular case. Moreover, the fact that coprimeness is a local property is proved and it is shown how coprimeness changes by performing Möbius transformations. [ABSTRACT FROM AUTHOR]

Matrices A of order n having entries in the field F ( x 1 , … , x n ) of rational functions over a field F and characteristic polynomial det ⁡ ( t I − A ) = t n + x 1 t n − 1 + ⋯ + x n − 1 t + x n are studied. It is known that such matrices are irreducible and have at least 2 n − 1 nonzero entries. Such matrices with exactly 2 n − 1 nonzero entries are called Ma–Zhan matrices. Conditions are given that imply that a Ma–Zhan matrix is similar via a monomial matrix to a generalized companion matrix (that is, a lower Hessenberg matrix with ones on its superdiagonal, and exactly one nonzero entry in each of its subdiagonals). Via the Ax–Grothendieck Theorem (respectively, its analog for the reals) these conditions are shown to hold for a family of matrices whose entries are complex (respectively, real) polynomials. [ABSTRACT FROM AUTHOR]

Let R be a unital associative algebra over a field K of characteristic zero, and let f be a multilinear polynomial of degree m over K . If m ≤ 3 , we prove that all traceless matrices can be written as the sum of two values of f evaluated over M n ( R ) with n ≥ 2 . If m = 4 , we prove the same result for n ≥ 3 . [ABSTRACT FROM AUTHOR]

We provide an algorithm for constructing strong ℓ -ifications of a given matrix polynomial P ( λ ) of degree d and size m × n using only the coefficients of the polynomial and the solution of linear systems of equations. A strong ℓ -ification of P ( λ ) is a matrix polynomial of degree ℓ having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial P ( λ ) . All explicit constructions of strong ℓ -ifications introduced so far in the literature have been limited to the case where ℓ divides d , though recent results on the inverse eigenstructure problem for matrix polynomials show that more general constructions are possible. Based on recent developments on dual polynomial minimal bases, we present a general construction of strong ℓ -ifications for wider choices of the degree ℓ , namely, when ℓ divides one of nd or md (and d ≥ ℓ ). In the case where ℓ divides nd (respectively, md ), the strong ℓ -ifications we construct allow us to easily recover the minimal indices of P ( λ ) . In particular, we show that they preserve the left (resp., right) minimal indices of P ( λ ) , and the right (resp., left) minimal indices of the ℓ -ification are the ones of P ( λ ) increased by d − ℓ (each). Moreover, in the particular case ℓ divides d , the new method provides a companion ℓ -ification that resembles very much the companion ℓ -ifications already known in the literature. [ABSTRACT FROM AUTHOR]

An elementary symplectic group w.r.t. an invertible alternating matrix is defined. It is shown that the group of symplectic transvections of a symplectic module coincides with this elementary symplectic group in the free case. Equality of orbit spaces of a unimodular element under the action of the linear group, symplectic group, and symplectic group w.r.t. an invertible alternating matrix is established. [ABSTRACT FROM AUTHOR]

Let R be a central simple algebra finite-dimensional over its center F of characteristic 0. We will show that every element of reduced trace 0 in R can be expressed as [ a , [ c , b ] ] + λ [ c , [ a , b ] ] for some a , b , c ∈ R where λ ≠ 0 , − 1 . In addition, let D be a division algebra satisfying the conditions above. We will also show that the set of values of any nonzero multilinear polynomial of degree at most three, with coefficients from the center F of D , evaluated on M k ( D ) , k ≥ 2 , contains all matrices of reduced trace 0. [ABSTRACT FROM AUTHOR]

An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a crucial feature of matrices in these classes. The behaviour of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Applications include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients. [ABSTRACT FROM PUBLISHER]

Minimal bases of rational vector spaces are a well-known and important tool in systems theory. If minimal bases for two subspaces of rational n -space are displayed as the rows of polynomial matrices Z 1 ( λ ) k × n and Z 2 ( λ ) m × n , respectively, then Z 1 and Z 2 are said to be dual minimal bases if the subspaces have complementary dimension, i.e., k + m = n , and Z 1 ( λ ) Z 2 T ( λ ) = 0 . In other words, each Z j ( λ ) provides a minimal basis for the nullspace of the other. It has long been known that for any dual minimal bases Z 1 ( λ ) and Z 2 ( λ ) , the row degree sums of Z 1 and Z 2 are the same. In this paper we show that this is the only constraint on the row degrees, thus characterizing the possible row degrees of dual minimal bases. The proof is constructive, making extensive use of a new class of sparse, structured polynomial matrices that we have baptized zigzag matrices. Another application of these polynomial zigzag matrices is the constructive solution of the following inverse problem for minimal indices: Given a list of left and right minimal indices and a desired degree d , does there exist a completely singular matrix polynomial (i.e., a matrix polynomial with no elementary divisors whatsoever) of degree d having exactly the prescribed minimal indices? We show that such a matrix polynomial exists if and only if d divides the sum of the minimal indices. The constructed realization is simple, and explicitly displays the desired minimal indices in a fashion analogous to the classical Kronecker canonical form of singular pencils. [ABSTRACT FROM AUTHOR]

Given a matrix polynomial P ( λ ) = ∑ i = 0 k λ i A i of degree k , where A i are n × n matrices with entries in a field F , the development of linearizations of P ( λ ) that preserve whatever structure P ( λ ) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P ( λ ) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of P ( λ ) available in the literature fall essentially into two classes: linearizations based on the so-called Fiedler pencils with repetition, which form a finite family, and a vector space of dimension k of block-symmetric pencils, called DL ( P ) , such that most of its pencils are linearizations. One drawback of the pencils in DL ( P ) is that none of them is a linearization when P ( λ ) is singular. In this paper we introduce new vector spaces of block-symmetric pencils, most of which are strong linearizations of P ( λ ) . The dimensions of these spaces are O ( n 2 ) , which, for n ≥ k , are much larger than the dimension of DL ( P ) . When k is odd, many of these vector spaces contain linearizations also when P ( λ ) is singular. The coefficients of the block-symmetric pencils in these new spaces can be easily constructed as k × k block-matrices whose n × n blocks are of the form 0, ± α I n , ± α A i , or arbitrary n × n matrices, where α is an arbitrary nonzero scalar. [ABSTRACT FROM AUTHOR]

We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as compound matrices, and on structural features including sparsity and symmetry. Results on the preservation of spectral information contained in elementary divisors, partial multiplicity sequences, invariant pairs, and minimal indices are presented. The effect on canonical forms such as Smith forms and local Smith forms, on relationships of strict equivalence and spectral equivalence, and on the property of being a linearization or quadratification are investigated. We show that many important transformations are special instances of Möbius transformations, and analyze a Möbius connection between alternating and palindromic matrix polynomials. Finally, the use of Möbius transformations in solving polynomial inverse eigenproblems is illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel and effective Euclidean algorithm for Laurent polynomial matrix extension (LPME), which is the key of the construction of perfect reconstruction filter banks (PRFBs). The algorithm gives an approach to the construction of PRFBs other than the dual-chain approach in the paper [5] . The developed algorithm can also be used in the applications where LPME plays a role. [ABSTRACT FROM AUTHOR]