| 1. |
- Friedlander, Benjaming, et al.
(författare)
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New Inversion Formulas for Matrices Classified in Terms of their Distance from Toeplitz Matrices
- 1979
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 27, s. 31-60
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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.
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| 2. |
- Hald, Ole H.
(författare)
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A converse to the Bauer-Fike theorem
- 1974
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 9, s. 267-274
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Tidskriftsartikel (refereegranskat)
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| 3. |
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| 4. |
- Akbari, Saieed, et al.
(författare)
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Chromatic number and clique number of subgraphs of regular graph of matrix algebras
- 2012
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 436:7, s. 2419-2424
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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.
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| 5. |
- Alexandersson, Per, et al.
(författare)
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Around multivariate Schmidt-Spitzer theorem
- 2014
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 446, s. 356-368
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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.
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| 6. |
- Altafi, Nasrin
(författare)
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Jordan types with small parts for Artinian Gorenstein algebras of codimension three
- 2022
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 646, s. 54-83
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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.
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| 7. |
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| 8. |
- Baricz, Árpád, et al.
(författare)
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The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants
- 2018
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 548, s. 259-272
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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.
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| 9. |
- Barros e Sa, Nuno, et al.
(författare)
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Families of complex Hadamard matrices
- 2013
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 438:7, s. 2929-2957
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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.
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| 10. |
- Berg, S., et al.
(författare)
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Ehrhart tensor polynomials
- 2018
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 539, s. 72-93
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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.
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