| 1. |
- Friedlander, Benjaming, et al.
(författare)
-
New Inversion Formulas for Matrices Classified in Terms of their Distance from Toeplitz Matrices
- 1979
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 27, s. 31-60
-
Tidskriftsartikel (refereegranskat)abstract
- The problem of solving linear equations, or equivalently of inverting matrices, arises in many fields. Efficient recursive algorithms for finding the inverses of Toeplitz or displacement-type matrices have been known for some time. By introducting a way of characterizing matrices in terms of their “distance” from being Toeplitz, a natural extension of these algorithms is obtained. Several new inversion formulas for the representation of the inverse of non-Toeplitz matrices are also presented.
|
|
| 2. |
- Hald, Ole H.
(författare)
-
A converse to the Bauer-Fike theorem
- 1974
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 9, s. 267-274
-
Tidskriftsartikel (refereegranskat)
|
|
| 3. |
|
|
| 4. |
- Akbari, Saieed, et al.
(författare)
-
Chromatic number and clique number of subgraphs of regular graph of matrix algebras
- 2012
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 436:7, s. 2419-2424
-
Tidskriftsartikel (refereegranskat)abstract
- Let R be a ring and X subset of R be a non-empty set. The regular graph of X, Gamma(X), is defined to be the graph with regular elements of X (non-zero divisors of X) as the set of vertices and two vertices are adjacent if their sum is a zero divisor. There is an interesting question posed in BCC22. For a field F, is the chromatic number of Gamma(GL(n)(F)) finite? In this paper, we show that if G is a soluble sub-group of GL(n)(F), then x (Gamma(G)) < infinity. Also, we show that for every field F, chi (Gamma(M-n(F))) = chi (Gamma(M-n(F(x)))), where x is an indeterminate. Finally, for every algebraically closed field F, we determine the maximum value of the clique number of Gamma(< A >), where < A > denotes the subgroup generated by A is an element of GL(n)(F). (C) 2011 Elsevier Inc. All rights reserved.
|
|
| 5. |
- Alexandersson, Per, et al.
(författare)
-
Around multivariate Schmidt-Spitzer theorem
- 2014
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 446, s. 356-368
-
Tidskriftsartikel (refereegranskat)abstract
- Given an arbitrary complex-valued infinite matrix $\infmatA=(a_{ij}),$$i=1,\dotsc,\infty;$ $j=1,\dotsc,\infty$ and a positive integer $n$ we introduce anaturally associated polynomial basis $\polybasis_\infmatA$ of$\C[x_0,\dotsc,x_n]$.We discuss some properties of the locus of common zeros of all polynomials in $\polybasis_A$ having a given degree $m$; the latter locus can beinterpreted as the spectrum of the $m\times (m+n)$-submatrix of $\infmatA$ formed by its $m$ first rows and$(m+n)$ first columns. We initiate the study of the asymptotics of these spectra when $m\to \infty$ inthe case when $\infmatA$ is a banded Toeplitz matrix.In particular, we present and partially prove a conjectural multivariate analogof the well-known Schmidt-Spitzer theorem which describes the spectral asymptotics for the sequence of principal minors of an arbitrarybanded Toeplitz matrix.Finally, we discuss relations between polynomial bases $\polybasis_\infmatA$ andmultivariate orthogonal polynomials.
|
|
| 6. |
- Altafi, Nasrin
(författare)
-
Jordan types with small parts for Artinian Gorenstein algebras of codimension three
- 2022
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 646, s. 54-83
-
Tidskriftsartikel (refereegranskat)abstract
- We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determined by at most three parameters.
|
|
| 7. |
|
|
| 8. |
- Baricz, Árpád, et al.
(författare)
-
The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants
- 2018
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 548, s. 259-272
-
Tidskriftsartikel (refereegranskat)abstract
- We derive a closed formula for the determinant of the Hankel matrix whose entries are given by sums of negative powers of the zeros of the regular Coulomb wave function. This new identity applied together with results of Grommer and Chebotarev allows us to prove a Hurwitz-type theorem about the zeros of the regular Coulomb wave function. As a particular case, we obtain a new proof of the classical Hurwitz's theorem from the theory of Bessel functions that is based on algebraic arguments. In addition, several Hankel determinants with entries given by the Rayleigh function and Bernoulli numbers are also evaluated.
|
|
| 9. |
- Barros e Sa, Nuno, et al.
(författare)
-
Families of complex Hadamard matrices
- 2013
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 438:7, s. 2929-2957
-
Tidskriftsartikel (refereegranskat)abstract
- What is the dimension of a smooth family of complex Hadamard matrices including the Fourier matrix? We address this problem with a power series expansion. Studying all dimensions up to 100 we find that the first order result is misleading unless the dimension is 6, or a power of a prime. In general the answer depends critically on the prime number decomposition of the dimension. Our results suggest that a general theory is possible. We discuss the case of dimension 12 in detail, and argue that the solution consists of two 13-dimensional families intersecting in a previously known 9-dimensional family. A precise conjecture for all dimensions equal to a prime times another prime squared is formulated.
|
|
| 10. |
- Berg, S., et al.
(författare)
-
Ehrhart tensor polynomials
- 2018
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 539, s. 72-93
-
Tidskriftsartikel (refereegranskat)abstract
- The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix cases, we give Pick-type formulas in terms of triangulations of a lattice polygon. As our main tool, we introduce hr-tensor polynomials, extending the notion of the Ehrhart h⁎-polynomial, and, for matrices, investigate their coefficients for positive semidefiniteness. In contrast to the usual h⁎-polynomial, the coefficients are in general not monotone with respect to inclusion. Nevertheless, we are able to prove positive semidefiniteness in dimension two. Based on computational results, we conjecture positive semidefiniteness of the coefficients in higher dimensions. Furthermore, we generalize Hibi's palindromic theorem for reflexive polytopes to hr-tensor polynomials and discuss possible future research directions.
|
|
| 11. |
- Bergqvist, Göran
(författare)
-
Curves and envelopes that bound the spectrum of a matrix
- 2018
-
Ingår i: Linear Algebra and its Applications. - : ELSEVIER SCIENCE INC. - 0024-3795 .- 1873-1856. ; 557
-
Tidskriftsartikel (refereegranskat)abstract
- A generalization of the method developed by Adam, Psarrakos and Tsatsomeros to find inequalities for the eigenvalues of a complex matrix A using knowledge of the largest eigenvalues of its Hermitian part H(A) is presented. The numerical range or field of values of A can be constructed as the intersection of half-planes determined by the largest eigenvalue of H(e(i theta) A.). Adam, Psarrakos and Tsatsomeros showed that using the two largest eigenvalues of H(A), the eigenvalues of A satisfy a cubic inequality and the envelope of such cubic curves defines a region in the complex plane smaller than the numerical range but still containing the spectrum of A. Here it is shown how using the three largest eigenvalues of H(A) or more, one obtains new inequalities for the eigenvalues of A and new envelope-type regions containing the spectrum of A. (C) 2018 Elsevier Inc. All rights reserved.
|
|
| 12. |
- Bergqvist, Göran, 1963-
(författare)
-
Exact probabilities for typical ranks of 2 × 2 × 2 and 3 × 3 × 2 tensors
- 2013
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 438:2, s. 663-667
-
Tidskriftsartikel (refereegranskat)abstract
- We show that the probability to be of rank 2 for a 2×2×2 tensor with elements from a standard normal distribution is π/4, and that the probability to be of rank 3 for a 3×3×2 tensor is 1/2. In the proof results on the expected number of real generalized eigenvalues of random matrices are applied. For n×n×2 tensors with n≥4 we also present some new aspects of their rank.
|
|
| 13. |
|
|
| 14. |
- Bock, Wolfgang, et al.
(författare)
-
An adjacency matrix perspective of talented monoids and Leavitt path algebras
- 2023
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 678, s. 295-316
-
Tidskriftsartikel (refereegranskat)abstract
- In this article we establish relationships between Leavitt path algebras, talented monoids and the adjacency matrices of the underlying graphs. We show that indeed the adjacency matrix generates in some sense the group action on the generators of the talented monoid. With the help of this, we deduce a form of the aperiodicity index of a graph via the talented monoid. We classify hereditary and saturated subsets via the adjacency matrix. This then translates to a correspondence between the composition series of the talented monoid and the so-called matrix composition series of the adjacency matrix. In addition, we discuss the number of cycles in a graph. In particular, we give an equivalent characterization of acyclic graphs via the adjacency matrix, the talented monoid and the Leavitt path algebra. Finally, we compute the number of linearly independent paths of certain length in the Leavitt path algebra via adjacency matrices.
|
|
| 15. |
- Bogoya, M., et al.
(författare)
-
Eigenvalue superposition for Toeplitz matrix-sequences with matrix order dependent symbols
- 2024
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 697, s. 487-527
-
Tidskriftsartikel (refereegranskat)abstract
- The eigenvalues of Toeplitz matrices T n ( f ) with a real-valued generating function f , satisfying some conditions and tracing out a simple loop over the interval [- pi, pi ], are known to admit an asymptotic expansion with the form lambda j ( T n ( f )) = f ( sigma j,n ) + c 1 ( sigma j,n ) h + c 2 ( sigma j,n ) h 2 + O ( h 3 ) , where h = 1 / ( n + 1), sigma j,n = pi jh , and c k are some bounded coefficients depending only on f . The numerical results presented in the literature suggest that the effective conditions for the expansion to hold are weaker and reduce to a fixed smoothness and to having only two intervals of monotonicity over [- pi, pi ]. In this article we investigate the superposition caused over this expansion, when considering the following linear combination lambda j ( T n ( f 0 ) + beta n, 1 T n ( f 1 ) + beta n, 2 T n ( f 2 ) ) , where beta n, 1 , beta n, 2 are certain constants depending on n and the generating functions f 0 , f 1 , f 2 are either simple loop or satisfy the weaker conditions mentioned before. We formally obtain an asymptotic expansion in this setting under simple -loop related assumptions, and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type including a multilevel setting. The problem is of concrete interest, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, and Isogeometric Analysis. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY -NC -ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).
|
|
| 16. |
- Boij, Mats, et al.
(författare)
-
A bound for the Waring rank of the determinant via syzygies
- 2020
-
Ingår i: Linear Algebra and its Applications. - : ELSEVIER SCIENCE INC. - 0024-3795 .- 1873-1856. ; 587, s. 195-214
-
Tidskriftsartikel (refereegranskat)abstract
- We show that the Waring rank of the 3 x 3 determinant, previously known to be between 14 and 18, is at least 15. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the symmetric cactus rank of the 3 x 3 permanent is at least 14.
|
|
| 17. |
- Bolten, Matthias, et al.
(författare)
-
Toeplitz momentary symbols : definition, results, and limitations in the spectral analysis of structured matrices
- 2022
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 651, s. 51-82
-
Tidskriftsartikel (refereegranskat)abstract
- A powerful tool for analyzing and approximating the singular values and eigenvalues of structured matrices is the theory of Generalized Locally Toeplitz (GLT) sequences. By the GLT theory one can derive a function, called the symbol, which describes the singular value or the eigenvalue distribution of the sequence, the latter under precise assumptions. However, for small values of the matrix-size of the considered sequence, the approximations may not be as good as it is desirable, since in the construction of the GLT symbol one disregards small norm and low-rank perturbations. On the other hand, Local Fourier Analysis (LFA) can be used to construct polynomial symbols in a similar manner for discretizations, where the geometric information is present, but the small norm perturbations are retained. The main focus of this paper is the introduction of the concept of sequence of "Toeplitz momentary symbols", associated with a given sequence of truncated Toeplitz-like matrices. We construct the symbol in the same way as in the GLT theory, but we keep the information of the small norm contributions. The low-rank contributions are still disregarded, and we give an idea on the reason why this is negligible in certain cases and why it is not in other cases, being aware that in presence of high nonnormality the same low-rank perturbation can produce a dramatic change in the eigenvalue distribution. Moreover, a difference with respect to the LFA symbols is that GLT symbols and Toeplitz momentary symbols are more general -just Lebesgue measurable -and are applicable to a larger class of matrices, while in the LFA setting only trigonometric polynomials are considered and more specifically those related to the approximation stencils. We show the applicability of the approach which leads to higher accuracy in some cases, when approximating the singular values and eigenvalues of Toeplitz-like matrices using Toeplitz momentary symbols, compared with the GLT symbol. Finally, since for many applications and their analysis it is often necessary to consider non-square Toeplitz matrices, we formalize and provide some useful definitions, applicable for non-square Toeplitz momentary symbols.
|
|
| 18. |
- Brändén, Petter
(författare)
-
Solutions to two problems on permanents
- 2012
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 436:1, s. 53-58
-
Tidskriftsartikel (refereegranskat)abstract
- In this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics. Both problems were recently discussed in Bapat's survey [2]. Bapat conjectured that certain quotients of permanents, which generalize symmetric function means, are concave. We prove this conjecture by using concavity properties of hyperbolic polynomials. Motivated by problems on random point processes, Shirai and Takahashi raised the problem: Determine all real numbers a for which the alpha-permanent (or alpha-determinant) is nonnegative for all positive semidefinite matrices. We give a complete solution to this problem by using recent results of Scott and Sokal on completely monotone functions. It turns out that the conjectured answer to the problem is false.
|
|
| 19. |
|
|
| 20. |
- Coco, Armando, et al.
(författare)
-
Spectral and norm estimates for matrix-sequences arising from a finite difference approximation of elliptic operators
- 2023
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 667, s. 10-43
-
Tidskriftsartikel (refereegranskat)abstract
- When approximating elliptic problems by using specialized approximation techniques, we obtain large structured matrices whose analysis provides information on the stability of the method. Here we provide spectral and norm estimates for matrix-sequences arising from the approximation of the Laplacian via ad hoc finite differences. The analysis involves several tools from matrix theory and in particular from the setting of Toeplitz operators and Generalized Locally Toeplitz matrix-sequences. Several numerical experiments are conducted, which confirm the correctness of the theoretical findings.
|
|
| 21. |
|
|
| 22. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
-
Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence
- 2015
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 469, s. 305-334
-
Tidskriftsartikel (refereegranskat)abstract
- We construct the Hasse diagrams G2 and G3 for the closure ordering on the sets of congruence classes of 2 × 2 and 3 × 3 complex matrices. In other words, we construct two directed graphs whose vertices are 2 × 2 or, respectively, 3 × 3 canonical matrices under congruence, and there is a directed path from A to B if and only if A can be transformed by an arbitrarily small perturbation to a matrix that is congruent to B. A bundle of matrices under congruence is defined as a set of square matrices A for which the pencils A + λAT belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of 2 × 2 or 3 × 3 matrices have the same properties with respect to perturbations. We construct the Hasse diagrams G2 B and G3 B for the closure ordering on the sets of congruence bundles of 2 × 2 and, respectively, 3 × 3 matrices. We find the isometry groups of 2 × 2 and 3 × 3 congruence canonical matrices.
|
|
| 23. |
|
|
| 24. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
-
Generalization of Roth's solvability criteria to systems of matrix equations
- 2017
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 527, s. 294-302
-
Tidskriftsartikel (refereegranskat)abstract
- W.E. Roth (1952) proved that the matrix equation AX - XB = C has a solution if and only if the matrices [Graphics] and [Graphics] are similar. A. Dmytryshyn and B. Kagstrom (2015) extended Roth's criterion to systems of matrix equations A(i)X(i')M(i) - (NiXi"Bi)-B-sigma i = Ci (i = 1,..., s) with unknown matrices X1,, X-t, in which every X-sigma is X, X-T, or X*. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations. (C) 2017 Elsevier Inc. All rights reserved.
|
|
| 25. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
-
Generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree
- 2017
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 535, s. 213-230
-
Tidskriftsartikel (refereegranskat)abstract
- The set POLd,rm×n of m×n complex matrix polynomials of grade d and (normal) rank at most r in a complex (d+1)mn dimensional space is studied. For r=1,...,min{m,n}−1, we show that POLd,rm×n is the union of the closures of the rd+1 sets of matrix polynomials with rank r, degree exactly d, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. r=min{m,n} and m≠n, we show that POLd,rm×n coincides with the closure of a single set of the polynomials with rank r, degree exactly d, and the described complete eigenstructure. These complete eigenstructures correspond to generic m×n matrix polynomials of grade d and rank at most r.
|
|
| 26. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
-
Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade
- 2018
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 536, s. 1-18
-
Tidskriftsartikel (refereegranskat)abstract
- We show that the set of m×m complex skew-symmetric matrix polynomials of odd grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m×m complex skew-symmetric matrix polynomials of odd grade d and rank at most 2r. In particular, this result includes the case of skew-symmetric matrix pencils (d=1).
|
|
| 27. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
-
Miniversal deformations of matrices of bilinear forms
- 2012
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 436:7, s. 2670-2700
-
Tidskriftsartikel (refereegranskat)abstract
- Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29–43] constructed miniversal deformations of square complex matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct miniversal deformations of matrices under congruence.
|
|
| 28. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
-
Miniversal deformations of matrices under *congruence and reducing transformations
- 2014
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 446:April, s. 388-420
-
Tidskriftsartikel (refereegranskat)abstract
- Arnold (1971) [1] constructed a miniversal deformation of a square complex matrix under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We give miniversal deformations of matrices of sesquilinear forms; that is, of square complex matrices under *congruence, and construct an analytic reducing transformation to a miniversal deformation. Analogous results for matrices under congruence were obtained by Dmytryshyn, Futorny, and Sergeichuk (2012) [11].
|
|
| 29. |
- Dmytryshyn, Andrii, 1986-
(författare)
-
Miniversal deformations of pairs of skew-symmetric matrices under congruence
- 2016
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 506, s. 506-534
-
Tidskriftsartikel (refereegranskat)abstract
- Miniversal deformations for pairs of skew-symmetric matrices under congruence are constructed. To be precise, for each such a pair (A, B) we provide a normal form with a minimal number of independent parameters to which all pairs of skew-symmetric matrices ((A) over tilde (,) (B) over tilde), close to (A, B) can be reduced by congruence transformation which smoothly depends on the entries of the matrices in the pair ((A) over tilde (,) (B) over tilde). An upper bound on the distance from such a miniversal deformation to (A, B) is derived too. We also present an example of using miniversal deformations for analyzing changes in the canonical structure information (i.e. eigenvalues and minimal indices) of skew-symmetric matrix pairs under perturbations.
|
|
| 30. |
- Dmytryshyn, Andrii, 1986-
(författare)
-
Miniversal deformations of pairs of symmetric matrices under congruence
- 2019
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 568, s. 84-105
-
Tidskriftsartikel (refereegranskat)abstract
- For each pair of complex symmetric matrices (A, B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices ((A) over tilde (B) over tilde), close to (A, B) can be reduced by congruence transformation that smoothly depends on the entries of (A ) over tilde and (B) over tilde. Such a normal form is called a miniversal deformation of (A, B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A, B) to its miniversal deformation.
|
|
| 31. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
-
Skew-symmetric matrix pencils : codimension counts and the solution of a pair of matrix equations
- 2013
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 438:8, s. 3375-3396
-
Tidskriftsartikel (refereegranskat)abstract
- The homogeneous system of matrix equations (X(T)A + AX, (XB)-B-T + BX) = (0, 0), where (A, B) is a pair of skew-symmetric matrices of the same size is considered: we establish the general solution and calculate the codimension of the orbit of (A, B) under congruence. These results will be useful in the development of the stratification theory for orbits of skew-symmetric matrix pencils.
|
|
| 32. |
- Dmytryshyn, Andrii, 1986-
(författare)
-
Structure preserving stratification of skew-symmetric matrix polynomials
- 2017
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 532, s. 266-286
-
Tidskriftsartikel (refereegranskat)abstract
- We study how elementary divisors and minimal indices of a skew-symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients. We investigate these changes qualitatively by constructing the stratifications (closure hierarchy graphs) of orbits and bundles for skew-symmetric linearizations. We also derive the necessary and sufficient conditions for the existence of a skew-symmetric matrix polynomial with prescribed degree, elementary divisors, and minimal indices.
|
|
| 33. |
- Dubsky, Brendan
(författare)
-
Classification of simple weight modules with finite-dimensional weight spaces over the Schrodinger algebra
- 2014
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 443, s. 204-214
-
Tidskriftsartikel (refereegranskat)abstract
- We classify simple weight modules with finite-dimensional weight spaces over the (centrally extended complex) Schrodinger algebra in (1 + 1)-dimensional space-time. Our arguments use the description of lowest weight modules by Dobrev, Doebner and Mrugalla; Mathieu's twisting functors and results of Wu and Zhu on dimensions of weight spaces in dense modules.
|
|
| 34. |
- Dubsky, Brendan Frisk, et al.
(författare)
-
Category O for the Schrodinger algebra
- 2014
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 460, s. 17-50
-
Tidskriftsartikel (refereegranskat)abstract
- We study category O for the (centrally extended) Schrodinger algebra. We determine the quivers for all blocks and relations for blocks of nonzero central charge. We also describe the quiver and relations for the finite dimensional part of O. We use this to determine the center of the universal enveloping algebra and annihilators of Verma modules. Finally, we classify primitive ideals of the universal enveloping algebra which intersect the center of the centrally extended Schrodinger algebra trivially.
|
|
| 35. |
- Ekström, Sven-Erik, et al.
(författare)
-
Eigenvalues and Eigenvectors of Tau Matrices with Applications to Markov Processes and Economics
- 2021
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 627, s. 41-71
-
Tidskriftsartikel (refereegranskat)abstract
- In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called τe,y algebra, a generalization of the well known τ algebra. We study the properties of eigenvalues and eigenvectors of the generator Tn,e,y of the τe,y algebra. In particular, we derive the asymptotics for the outliers of Tn,e,y and the associated eigenvectors; we obtain equations for the eigenvalues of Tn,e,y, which provide also the eigenvectors of Tn,e,y; and we compute the full eigendecomposition of Tn,e,y in the specific case ey=1. We also present applications of our results in the context of queuing models, random walks, and diffusion processes, with a special attention to their implications in the study of wealth/income inequality and portfolio dynamics.
|
|
| 36. |
- Eldén, Lars
(författare)
-
Computing Frechet derivatives in partial least squares regression
- 2015
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 473, s. 316-338
-
Tidskriftsartikel (refereegranskat)abstract
- Partial least squares is a common technique for multivariate regression. The pro- cedure is recursive and in each step basis vectors are computed for the explaining variables and the solution vectors. A linear model is fitted by projection onto the span of the basis vectors. The procedure is mathematically equivalent to Golub-Kahan bidiagonalization, which is a Krylov method, and which is equiv- alent to a pair of matrix factorizations. The vectors of regression coefficients and prediction are non-linear functions of the right hand side. An algorithm for computing the Frechet derivatives of these functions is derived, based on perturbation theory for the matrix factorizations. From the Frechet derivative of the prediction vector one can compute the number of degrees of freedom, which can be used as a stopping criterion for the recursion. A few numerical examples are given.
|
|
| 37. |
- Ernst, Thomas
(författare)
-
An umbral approach to find q-analogues of matrix formulas
- 2013
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 439:4, s. 1167-1182
-
Tidskriftsartikel (refereegranskat)abstract
- A general introduction is given to the logarithmic q-analogue formulation of mathematical expressions with a special focus on its use for matrix calculations. The fundamental definitions relevant to q-analogues of mathematical objects are given and form the basis for matrix formulations in the paper. The umbral approach is used to find q-analogues of significant matrices. Finally, as an explicit example, a new formula for q-Cauchy-Vandermonde determinant containing matrix elements equal to q-numbers introduced by Ward is proved by using a new type of q-Stirling numbers together with Lagrange interpolation in Z(q).
|
|
| 38. |
- Ernst, Thomas
(författare)
-
On several q-special matrices, including the q-Bernoulli and q-Euler matrices
- 2018
-
Ingår i: Linear Algebra and its Applications. - : ELSEVIER SCIENCE INC. - 0024-3795 .- 1873-1856. ; 542, s. 422-440
-
Tidskriftsartikel (refereegranskat)abstract
- In the spirit of our earlier articles [12,14,11], and our work [13], we define two dual q-Bernoulli polynomials, with corresponding vector and matrix forms. Following Aceto Trigiante [1], the q-L matrix, the indefinite q-integral of the q-Pascal matrix is the link between the q-Cauchy and the q-Bernoulli matrix. The q-analogue of the Bernoulli complementary argument theorem can be expressed in matrix form through the diagonal An matrix. For the q-Euler polynomials corresponding results are obtained. The umbral calculus for generating functions of q-Appell polynomials is shown to be equivalent to a transform method, which maps polynomials to matrices, a true q-analogue of Arponen [6]. This is manifested by the Vein [21] matrix, which occurs as the transform of the q-difference operator. The Aceto Trigiante shifted q-Bernoulli matrix has a simple connection to the q-Bernoulli Arponen matrix through the q-Pascal matrix. We reintroduce certain q-Stirling numbers is an element of 7L(q) from [12], which will be needed for the polynomial matrix definitions.
|
|
| 39. |
- Fanizza, Giovanna, et al.
(författare)
-
Passivity-preserving model reduction by analytic interpolation
- 2007
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 425:2-3, s. 608-633
-
Tidskriftsartikel (refereegranskat)abstract
- Antoulas and Sorensen have recently proposed a passivity-preserving model-reduction method of linear systems based on Krylov projections. The idea is to approximate a positive-real rational transfer function with one of lower degree. The method is based on an observation by Antoulas (in the single-input/single-output case) that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros, then the approximant is also positive real. However, this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by Byrnes, Georgiou and Lindquist, namely the maximum-entropy (central) solution. By tuning the interpolation points and the spectral zeros, as prescribed by this theory, one is able to obtain considerably better reduced-order models. We also show that, in the multi-input/multi-output case, Sorensen's algorithm actually amounts to tangential Nevanlinna-Pick interpolation.
|
|
| 40. |
- Fasi, Massimiliano, 1989-, et al.
(författare)
-
Sampling the eigenvalues of random orthogonal and unitary matrices
- 2021
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 620, s. 297-321
-
Tidskriftsartikel (refereegranskat)abstract
- We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such matrices, and then computes their eigenvalues with a tailored core-chasing algorithm. This approach requires a number of floating-point operations that is quadratic in the order of the matrix being sampled, and can be adapted to other matrix groups. In particular, we explain how it can be used to sample the Haar measure over the special orthogonal and unitary groups and the conditional probability distribution obtained by requiring the determinant of the sampled matrix be a given complex number on the complex unit circle.
|
|
| 41. |
- Gernandt, Hannes, et al.
(författare)
-
A Calderón type inverse problem for tree graphs
- 2022
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 646, s. 29-42
-
Tidskriftsartikel (refereegranskat)abstract
- We study the inverse problem of recovering a tree graph together with the weights on its edges (equivalently a metric tree) from the knowledge of the Dirichlet-to-Neumann matrix associated with the Laplacian. We prove an explicit formula which relates this matrix to the pairwise weighted distances of the leaves of the tree and, thus, allows to recover the weighted tree. This result can be viewed as a counterpart of the Calderón problem in the analysis of PDEs. In contrast to earlier results on inverse problems for metric graphs, we only assume knowledge of the Dirichlet-to-Neumann matrix for a fixed energy, not of a whole matrix-valued function.
|
|
| 42. |
|
|
| 43. |
- Jarlebring, Elias
(författare)
-
Convergence factors of Newton methods for nonlinear eigenvalue problems
- 2012
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 436:10, s. 3943-3953
-
Tidskriftsartikel (refereegranskat)abstract
- Consider a complex sequence convergent to λ∗∈C with order p∈N. The convergence factor is typically defined as the fraction ck:=(λk+1-λ∗)/(λk-λ∗)p in the limit k→∞. In this paper, we prove formulas characterizing ck in the limit k→∞ for two different Newton-type methods for nonlinear eigenvalue problems. The formulas are expressed in terms of the left and right eigenvectors.The two treated methods are called the method of successive linear problems (MSLP) and augmented Newton and are widely used in the literature. We prove several explicit formulas for ck for both methods. Formulas for both methods are found for simple as well as double eigenvalues. In some cases, we observe in examples that the limit ck as k→∞ does not exist. For cases where this limit does not appear to exist, we prove other limiting expressions such that a characterization of ck in the limit is still possible.
|
|
| 44. |
- Jarlebring, Elias, et al.
(författare)
-
Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations
- 2009
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 431:3-4, s. 369-380
-
Tidskriftsartikel (refereegranskat)abstract
- Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called matrix pencil methods, the general ideas used as well as the proofs differ considerably. Moreover, the available theory hardly reveals the relations between the different methods. In this work, a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem: the polynomial two-parameter eigenvalue problem, of which the quadratic two-parameter eigenvalue problem is a special case. This framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods. We also recognize a few new matrix pencil variants to determine DDE stability. Finally, the recognition of the new types of eigenvalue problem opens a door to efficient computation of DDE stability. (C) 2009 Elsevier Inc. All rights reserved.
|
|
| 45. |
- Johansson, Stefan, et al.
(författare)
-
Stratification of full rank polynomial matrices
- 2013
-
Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 439:4, s. 1062-1090
-
Tidskriftsartikel (refereegranskat)abstract
- We show that perturbations of polynomial matrices of full normal-rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices.It is proved that a full normal-rank polynomial matrix has the same structural elements asits right (or left) linearization. Furthermore, the linearized pencil has a special structurethat can be taken into account when studying its stratification. This yields constraintson the set of achievable eigenstructures. We explicitly show which these constraints are.These results allow us to derive necessary and sufficient conditions for cover relationsbetween two orbits or bundles of the linearization of full normal-rank polynomial matrices.The stratification rules are applied to and illustrated on two artificial polynomial matricesand a half-car passive suspension system with four degrees of freedom.
|
|
| 46. |
- Karlsson, Bengt R.
(författare)
-
BCCB complex Hadamard matrices of order 9, and MUBs
- 2016
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 504, s. 309-324
-
Tidskriftsartikel (refereegranskat)abstract
- A new type of complex Hadamard matrices of order 9 are constructed. The studied matrices are symmetric, block circulant with circulant blocks (BCCB) and form an until now unknown non-reducible and non-affine two-parameter orbit. Several suborbits are identified, including a one parameter intersection with the Fourier orbit F-9((4)). The defect of this new type of Hadamard matrices is observed to vary, from a generic value 2 to the anomalous values 4 and 10 for some sub-orbits, and to 12 and 16 for some single matrices. The latter matrices are shown to be related to complete sets of MUBs in dimension 9.
|
|
| 47. |
- Karlsson, Bengt R.
(författare)
-
H-2-reducible complex Hadamard matrices of order 6
- 2011
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 434:1, s. 239-246
-
Tidskriftsartikel (refereegranskat)abstract
- Complex Hadamard matrices H of order 6 are characterized in a novel manner, according to the presence/absence of order 2 Hadamard submatrices. It is shown that if there exists one such submatrix. H is equivalent to a Hadamard matrix where all the nine submatrices are Hadamard. The ensuing subset of H-2-reducible complex Hadamard matrices is more general than might be thought, and, significantly, includes all the up till now described (one- and two-parameter) families of order 6. A known, isolated matrix, and most numerically generated matrices, fall outside the subset.
|
|
| 48. |
- Karlsson, Bengt R.
(författare)
-
Three-parameter complex Hadamard matrices of order 6
- 2011
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 434:1, s. 247-258
-
Tidskriftsartikel (refereegranskat)abstract
- A three-parameter family of complex Hadamard matrices of order 6 is presented. It significantly extends the set of closed form complex Hadamard matrices of this order, and in particular contains all previously described one- and two-parameter families as subfamilies.
|
|
| 49. |
- Konstantopoulos, Takis
(författare)
-
A multilinear algebra proof of the Cauchy-Binet formula and a multilinear version of Parseval's identity
- 2013
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 439:9, s. 2651-2658
-
Tidskriftsartikel (refereegranskat)abstract
- We prove the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity not involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of the classical Parseval identity.
|
|
| 50. |
- Krutov, Andrey, et al.
(författare)
-
Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2
- 2022
-
Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 649, s. 1-21
-
Tidskriftsartikel (refereegranskat)abstract
- As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2.We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or 0|1, or 1|1; it is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic ≠2, see arXiv:1806.05505).We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2.We also recall examples of multiple NISes on simple Lie algebras over non-closed fields.
|
|
