| 1. |
- Friedlander, Benjaming, et al.
(author)
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New Inversion Formulas for Matrices Classified in Terms of their Distance from Toeplitz Matrices
- 1979
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In: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 27, s. 31-60
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Journal article (peer-reviewed)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. |
- Gulliksson, Mårten, et al.
(author)
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Perturbation bounds for constrained and weighted least squares problems
- 2002
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In: Linear Algebra and Its Applications. - 0024-3795. ; 349, s. 221-232
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Journal article (peer-reviewed)abstract
- We derive perturbation bounds for the constrained and weighted linear least squares (LS) problems. Both the full rank and rank-deficient cases are considered. The analysis generalizes some results of earlier works.
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| 3. |
- Hald, Ole H.
(author)
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A converse to the Bauer-Fike theorem
- 1974
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In: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 9, s. 267-274
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Journal article (peer-reviewed)
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| 4. |
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| 5. |
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| 6. |
- Akbari, Saieed, et al.
(author)
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Chromatic number and clique number of subgraphs of regular graph of matrix algebras
- 2012
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In: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 436:7, s. 2419-2424
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Journal article (peer-reviewed)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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| 7. |
- Alexandersson, Per, et al.
(author)
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Around multivariate Schmidt-Spitzer theorem
- 2014
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In: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 446, s. 356-368
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Journal article (peer-reviewed)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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| 8. |
- Altafi, Nasrin
(author)
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Jordan types with small parts for Artinian Gorenstein algebras of codimension three
- 2022
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In: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 646, s. 54-83
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Journal article (peer-reviewed)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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| 9. |
- Andreotti, Eleonora, 1988, et al.
(author)
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Measuring stability of spectral clustering
- 2019
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In: Linear Algebra and Its Applications. - : Elsevier BV. - 0024-3795. ; 610, s. 673-697
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Journal article (peer-reviewed)abstract
- As an indicator of the stability of spectral clustering of an undirected weighted graph into k clusters, the kth spectral gap of the graph Laplacian is often considered. The k-th spectral gap is characterized here as an unstructured distance to ambiguity, namely as the minimal distance of the Laplacian to arbitrary symmetric matrices with vanishing kth spectral gap. As a more appropriate measure of stability, the structured distance to ambiguity of the k-clustering is introduced as the minimal distance of the Laplacian to Laplacians of the same graph with weights that are perturbed such that the k-th spectral gap vanishes. To compute a solution to this matrix nearness problem, a two-level iterative algorithm is proposed that uses a constrained gradient system of matrix differential equations in the inner iteration and a one-dimensional optimization of the perturbation size in the outer iteration. The structured and unstructured distances to ambiguity are compared on some example graphs. The numerical experiments show, in particular, that selecting the number k of clusters according to the criterion of maximal stability can lead to different results for the structured and unstructured stability indicators.
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