Author
LeRoy B. Beasley
Other affiliations: SRI International, University of British Columbia
Bio: LeRoy B. Beasley is an academic researcher from Utah State University. The author has contributed to research in topics: Rank (linear algebra) & Matrix (mathematics). The author has an hindex of 19, co-authored 126 publications receiving 1213 citations. Previous affiliations of LeRoy B. Beasley include SRI International & University of British Columbia.
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TL;DR: In this paper, the authors consider a function on a matrix space M which commutes with a linear operator on M to preserve certain properties associated with positive semidefinite matrices.
Abstract: This chapter consists of three sections of miscellaneous results. Each is self contained with its own abstract, but the references are in one group at the very end of the chapter. The first section considers matrices over semirings. The second concerns a function on a matrix space M which commutes with a linear operator on M. The third considers linear operators preserving certain properties associated with positive semidefinite matrices.
102 citations
TL;DR: In this article, the authors studied the extent to which certain theorems on linear operators on field-valued matrices carry over to linear operations on Boolean matrices and obtained analogues and near analogues of several such theoresms.
79 citations
TL;DR: In this article, the authors considered the set of m × n nonnegative real matrices and defined the nonnegative rank of a matrix A to be the minimum k such that A = BC where B is m × k and C is k × n.
54 citations
TL;DR: In this article, it was shown that the dimension of a subspace of m×n matrices over any field with at least k+1 elements whose nonzero elements all have rank k is at most max(m,n).
51 citations
TL;DR: This paper investigates the largest value of r for which the column rank and semiring rank of all m × n matrices over a given semiring are both r .
49 citations
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TL;DR: In this article, the structures of commutativity-preserving mappings, Lie isomorphisms, and Lie derivations of certain prime rings are derived for all x ∈ R.
Abstract: Biadditive mappings B: R × R → R where R is a prime ring with certain additional properties, satisfying B(x, x)x = xB(x, x) for all x ∈ R, are characterized. As an application we determine the structures of commutativity-preserving mappings, Lie isomorphisms, and Lie derivations of certain prime rings
339 citations
TL;DR: Linear preserver problems as discussed by the authors concern the characterization of linear operators on matrix spaces that leave certain functions, subsets, relations, etc., invariant, and a great deal of effort has been devoted to the study of this type of question since then.
246 citations
TL;DR: This article describes some techniques, outlines a few proofs, and discusses some exceptional results of linear preserver problems, an active research area in matrix and operator theory.
Abstract: Linear preserver problems is an active research area in matrix and operator theory. These problems involve certain linear operators on spaces of matrices or operators. We give a general introduction to the subject in this article. In the first three sections, we discuss motivation, results, and problems. In the last three sections, we describe some techniques, outline a few proofs, and discuss some exceptional results. 1. EXAMPLES AND TYPICAL PROBLEMS. Let Mm,n be the set of m × n complex matrices, and let Mn = Mn,n. Suppose that M, N ∈ Mn satisfy det( MN ) = 1. Then the mapping φ : Mn → Mn given by
233 citations
TL;DR: In this article, the authors survey the development of the theory of commuting maps and their applications, including derivations, commuting additive maps, commuting traces of multiadditive maps, various generalizations of the notion of a commuting map, and applications of results on commuting maps to Lie theory.
Abstract: A map $f$ on a ring $\cal A$ is said to be commuting if $f(x)$ commutes with $x$ for every $x\in \cal A$. The paper surveys the development of the theory of commuting maps and their applications. The following topics are discussed: commuting derivations, commuting additive maps, commuting traces of multiadditive maps, various generalizations of the notion of a commuting map, and applications of results on commuting maps to different areas, in particular to Lie theory.
193 citations