Reciprocity and Antireciprocity Criteria by Elementary Matrix Techniques

Abstract We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q(Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ġ ) – xI (x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.

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Extending Maxima Capabilities for Performing Elementary Matrix Operations

Computational Science and Its Applications – ICCSA 2020 - Lecture Notes in Computer Science ◽

10.1007/978-3-030-58799-4_30 ◽

2020 ◽

pp. 406-420

Author(s):

Karina F. M. Castillo-Labán ◽

Robert Ipanaqué-Chero

Keyword(s):

Elementary Matrix ◽

Matrix Operations

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Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

Special Matrices ◽

10.1515/spma-2016-0010 ◽

2016 ◽

Vol 4 (1) ◽

Cited By ~ 2

Author(s):

Mika Mattila ◽

Pentti Haukkanen

Keyword(s):

Real Numbers ◽

Matrix Methods ◽

Elementary Matrix

AbstractLet T = {z1, z2, . . . , zn} be a finite multiset of real numbers, where z1 ≤ z2 ≤ · · · ≤ zn. The purpose of this article is to study the different properties of MIN and MAX matrices of the set T with min(zi , zj) and max(zi , zj) as their ij entries, respectively.We are going to do this by interpreting these matrices as so-called meet and join matrices and by applying some known results for meet and join matrices. Once the theorems are found with the aid of advanced methods, we also consider whether it would be possible to prove these same results by using elementary matrix methods only. In many cases the answer is positive.

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Another proof of a theorem on difference sets

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100041566 ◽

1967 ◽

Vol 63 (3) ◽

pp. 595-596

Author(s):

D. L. Yates

Keyword(s):

Cyclic Group ◽

Elementary Proof ◽

Existence Theorems ◽

Difference Sets ◽

Difference Set ◽

En Bloc ◽

Elementary Matrix

Multipliers of a difference set are of great importance in existence theorems, since they enable us to reject many configurations en bloc. (For a description of such theorems, see Mann (1).) The following theorem, which determines those cyclic group difference sets for which −1 is a multiplier, has been proved before by different methods (see, for example, Yamamoto(2) and Johnsen(3); and a more elementary matrix proof by Brualdi(4)) but the following ‘elementary’ proof may be of interest.

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Elementary matrix reduction over Bézout domains

Journal of Algebra and Its Applications ◽

10.1142/s021949881950141x ◽

2019 ◽

Vol 18 (08) ◽

pp. 1950141

Author(s):

Huanyin Chen ◽

Marjan Sheibani Abdolyousefi

Keyword(s):

Elementary Divisor ◽

Bezout Domain ◽

Elementary Matrix ◽

Matrix Reduction ◽

Elementary Divisor Ring ◽

Bezout Domains ◽

Diagonal Reduction

A ring [Formula: see text] is an elementary divisor ring if every matrix over [Formula: see text] admits a diagonal reduction. If [Formula: see text] is an elementary divisor domain, we prove that [Formula: see text] is a Bézout duo-domain if and only if for any [Formula: see text], [Formula: see text] such that [Formula: see text]. We explore certain stable-like conditions on a Bézout domain under which it is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.

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