2012
DOI: 10.1016/j.laa.2012.06.042
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Standard orthogonal vectors in semilinear spaces and their applications

Abstract: This paper investigates the standard orthogonal vectors in semilinear spaces of n-dimensional vectors over commutative zerosumfree semirings. First, we discuss some characterizations of standard orthogonal vectors. Then as applications, we obtain some necessary and sufficient conditions that a set of vectors is a basis of a semilinear subspace which is generated by standard orthogonal vectors, prove that a set of linearly independent nonstandard orthogonal vectors cannot be orthogonalized if it has at least tw… Show more

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Cited by 12 publications
(3 citation statements)
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References 15 publications
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“…Semimodules and semirings were studied by several authors, let us mention at least the papers [8,9,12,13]. Since these concepts are defined differently by the different authors, for the reader's convenience we provide the following definition.…”
Section: Semimodules Over Semiringsmentioning
confidence: 99%
“…Semimodules and semirings were studied by several authors, let us mention at least the papers [8,9,12,13]. Since these concepts are defined differently by the different authors, for the reader's convenience we provide the following definition.…”
Section: Semimodules Over Semiringsmentioning
confidence: 99%
“…Semimodules and semirings were studied by several authors, let us mention at least the papers [6], [10], [11] and [12]. Since these concepts are defined differently by the different authors, for the reader's convenience we provide the following definition.…”
Section: Semimodules Over Semiringsmentioning
confidence: 99%
“…In 2011, Shu and Wang showed some necessary and sufficient conditions that each basis has the same number of elements over commutative zerosumfree semirings and proved that a set of vectors is a basis if and only if they are standard orthogonal (see [15]). In 2012, Shu and Wang showed that a set of linearly independent non standard orthogonal vectors can not be orthogonalized if it has at least two nonzero vectors, and proved that the analog of the Kronecker-Capelli theorem was valid for systems of equations when the column vectors of coefficient matrix was standard orthogonal(see [16]). It is obvious that the problem is still open if the column vectors of coefficient matrix is not standard orthogonal.…”
Section: Introductionmentioning
confidence: 99%