| 1. |
- 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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| 2. |
- 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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| 3. |
- 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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| 4. |
- Bergqvist, Göran, 1963-
(författare)
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Exact probabilities for typical ranks of 2 × 2 × 2 and 3 × 3 × 2 tensors
- 2013
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 438:2, s. 663-667
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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.
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| 5. |
- Brändén, Petter
(författare)
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Solutions to two problems on permanents
- 2012
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 436:1, s. 53-58
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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.
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| 6. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
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Miniversal deformations of matrices of bilinear forms
- 2012
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 436:7, s. 2670-2700
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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.
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| 7. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
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Miniversal deformations of matrices under *congruence and reducing transformations
- 2014
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 446:April, s. 388-420
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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].
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| 8. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
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Skew-symmetric matrix pencils : codimension counts and the solution of a pair of matrix equations
- 2013
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 438:8, s. 3375-3396
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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.
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| 9. |
- Dubsky, Brendan
(författare)
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Classification of simple weight modules with finite-dimensional weight spaces over the Schrodinger algebra
- 2014
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 443, s. 204-214
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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.
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| 10. |
- Dubsky, Brendan Frisk, et al.
(författare)
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Category O for the Schrodinger algebra
- 2014
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 460, s. 17-50
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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.
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