Abstract In this article, we consider the relationships between walks in a signed graph G˙ dot{G} and its eigenvalues, with a particular focus on the largest absolute value of its eigenvalues ρ(G˙) rho left(dot{G}) , known as the spectral radius. Among other results, we derive a sequence of lower bounds for ρ(G˙) rho left(dot{G}) expressed in terms of walks or closed walks. We also prove that ρ(G˙) rho left(dot{G}) attains the spectral radius of the underlying graph G G if and only if G˙ dot{G} is switching equivalent to G G or its negation. It is proved that the length k k of the shortest negative cycle in G˙ dot{G} and the number of such cycles are determined by the spectrum of G˙ dot{G} and the spectrum of G G . Finally, a relation between k k and characteristic polynomials of G˙ dot{G} and G G is established.
{"title":"Walks and eigenvalues of signed graphs","authors":"Zoran Stanić","doi":"10.1515/spma-2023-0104","DOIUrl":"https://doi.org/10.1515/spma-2023-0104","url":null,"abstract":"Abstract In this article, we consider the relationships between walks in a signed graph <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mover accent=\"true\"> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:math> dot{G} and its eigenvalues, with a particular focus on the largest absolute value of its eigenvalues <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ρ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mover accent=\"true\"> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> rho left(dot{G}) , known as the spectral radius. Among other results, we derive a sequence of lower bounds for <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ρ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mover accent=\"true\"> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> rho left(dot{G}) expressed in terms of walks or closed walks. We also prove that <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ρ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mover accent=\"true\"> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> rho left(dot{G}) attains the spectral radius of the underlying graph <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> G if and only if <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mover accent=\"true\"> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:math> dot{G} is switching equivalent to <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> G or its negation. It is proved that the length <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> k of the shortest negative cycle in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mover accent=\"true\"> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:math> dot{G} and the number of such cycles are determined by the spectrum of <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mover accent=\"true\"> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:math> dot{G} and the spectrum of <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> G . Finally, a relation between <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> k and characteristic polynomials of <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mover accent=\"true\"> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:math> dot{G} and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> G is established.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135699236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The solvability of several new constrained quaternion matrix equations is investigated, and their unique solutions are presented in terms of the weighted MPD inverse and weighted DMP inverse of suitable matrices. It is interesting to consider some exceptional cases of these new equations and corresponding solutions. Determinantal representations for the solutions of the equations as mentioned earlier are established as sums of appropriate minors. In order to illustrate the obtained results, a numerical example is shown.
{"title":"W-MPD–N-DMP-solutions of constrained quaternion matrix equations","authors":"Ivan Kyrchei, D. Mosić, P. Stanimirović","doi":"10.1515/spma-2022-0183","DOIUrl":"https://doi.org/10.1515/spma-2022-0183","url":null,"abstract":"Abstract The solvability of several new constrained quaternion matrix equations is investigated, and their unique solutions are presented in terms of the weighted MPD inverse and weighted DMP inverse of suitable matrices. It is interesting to consider some exceptional cases of these new equations and corresponding solutions. Determinantal representations for the solutions of the equations as mentioned earlier are established as sums of appropriate minors. In order to illustrate the obtained results, a numerical example is shown.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48498225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract For a bipartite graph, the complete adjacency matrix is not necessary to display its adjacency information. In 1985, Godsil used a smaller size matrix to represent this, known as the bipartite adjacency matrix. Recently, the bipartite distance matrix of a tree with perfect matching was introduced as a concept similar to the bipartite adjacency matrix. It has been observed that these matrices are nonsingular, and a combinatorial formula for their determinants has been derived. In this article, we provide a combinatorial description of the inverse of the bipartite distance matrix and establish identities similar to some well-known identities. The study leads us to an unexpected generalization of the usual Laplacian matrix of a tree. This generalized Laplacian matrix, which we call the bipartite Laplacian matrix, is usually not symmetric, but it shares many properties with the usual Laplacian matrix. In addition, we study some of the fundamental properties of the bipartite Laplacian matrix and compare them with those of the usual Laplacian matrix.
{"title":"The bipartite Laplacian matrix of a nonsingular tree","authors":"R. Bapat, Rakesh Jana, S. Pati","doi":"10.1515/spma-2023-0102","DOIUrl":"https://doi.org/10.1515/spma-2023-0102","url":null,"abstract":"Abstract For a bipartite graph, the complete adjacency matrix is not necessary to display its adjacency information. In 1985, Godsil used a smaller size matrix to represent this, known as the bipartite adjacency matrix. Recently, the bipartite distance matrix of a tree with perfect matching was introduced as a concept similar to the bipartite adjacency matrix. It has been observed that these matrices are nonsingular, and a combinatorial formula for their determinants has been derived. In this article, we provide a combinatorial description of the inverse of the bipartite distance matrix and establish identities similar to some well-known identities. The study leads us to an unexpected generalization of the usual Laplacian matrix of a tree. This generalized Laplacian matrix, which we call the bipartite Laplacian matrix, is usually not symmetric, but it shares many properties with the usual Laplacian matrix. In addition, we study some of the fundamental properties of the bipartite Laplacian matrix and compare them with those of the usual Laplacian matrix.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41449725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ilias S. Kotsireas, Christoph Koutschan, Dursun A. Bulutoglu, David M. Arquette, Jonathan S. Turner, Kenneth J. Ryan
Abstract By assuming a type of balance for length ℓ = 87 ell =87 and nontrivial subgroups of multiplier groups of Legendre pairs (LPs) for length ℓ = 85 ell =85 , we find LPs of these lengths. We then study the power spectral density (PSD) values of m m compressions of LPs of length 5 m 5m . We also formulate a conjecture for LPs of lengths ℓ ≡ 0 ell equiv 0 (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range ≤ 200 le 200 for which the existence question of LPs remains unsolved from 12 to 10.
{"title":"Legendre pairs of lengths <i>ℓ</i> ≡ 0 (mod 5)","authors":"Ilias S. Kotsireas, Christoph Koutschan, Dursun A. Bulutoglu, David M. Arquette, Jonathan S. Turner, Kenneth J. Ryan","doi":"10.1515/spma-2023-0105","DOIUrl":"https://doi.org/10.1515/spma-2023-0105","url":null,"abstract":"Abstract By assuming a type of balance for length ℓ = 87 ell =87 and nontrivial subgroups of multiplier groups of Legendre pairs (LPs) for length ℓ = 85 ell =85 , we find LPs of these lengths. We then study the power spectral density (PSD) values of m m compressions of LPs of length 5 m 5m . We also formulate a conjecture for LPs of lengths ℓ ≡ 0 ell equiv 0 (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range ≤ 200 le 200 for which the existence question of LPs remains unsolved from 12 to 10.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135611402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Edges in the graph associated with a square matrix over a field may be classified as to how their removal affects the multiplicity of an identified eigenvalue. There are five possibilities: + 2 +2 (2-Parter); + 1 +1 (Parter); no change (neutral); − 1 -1 (downer); and − 2 -2 (2-downer). Especially, it is known that 2-downer edges for an eigenvalue comprise cycles in the graph. We investigate the effect for the statuses of other edges or vertices by removing a 2-downer edge. Then, we investigate the change in the multiplicity of an eigenvalue by removing a cut 2-downer edge triangle.
{"title":"The effect of removing a 2-downer edge or a cut 2-downer edge triangle for an eigenvalue","authors":"K. Toyonaga","doi":"10.1515/spma-2022-0186","DOIUrl":"https://doi.org/10.1515/spma-2022-0186","url":null,"abstract":"Abstract Edges in the graph associated with a square matrix over a field may be classified as to how their removal affects the multiplicity of an identified eigenvalue. There are five possibilities: + 2 +2 (2-Parter); + 1 +1 (Parter); no change (neutral); − 1 -1 (downer); and − 2 -2 (2-downer). Especially, it is known that 2-downer edges for an eigenvalue comprise cycles in the graph. We investigate the effect for the statuses of other edges or vertices by removing a 2-downer edge. Then, we investigate the change in the multiplicity of an eigenvalue by removing a cut 2-downer edge triangle.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41395607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Denote by σ n {sigma }_{n} the n-th Stirling polynomial in the sense of the well-known book Concrete Mathematics by Graham, Knuth and Patashnik. We show that there exist developments x σ n ( x ) = ∑ j = 0 n ( 2 j j ! ) − 1 q n − j ( j ) x j x{sigma }_{n}left(x)={sum }_{j=0}^{n}{left({2}^{j}j!)}^{-1}{q}_{n-j}left(j){x}^{j} with polynomials q j {q}_{j} of degree j . j. We deduce from this the polynomial identities ∑ a + b + c + d = n ( − 1 ) d ( x − 2 a − 2 b ) 3 n − s − a − c a ! b ! c ! d ! ( 3 n − s − a − c ) ! = 0 , for s ∈ Z ≥ 1 , sum _{a+b+c+d=n}{left(-1)}^{d}frac{{left(x-2a-2b)}^{3n-s-a-c}}{a!b!c!d!left(3n-s-a-c)!}=0,hspace{1.0em}hspace{0.1em}text{for}hspace{0.1em}hspace{0.33em}sin {{mathbb{Z}}}_{ge 1}, found in an attempt to find a simpler formula for the density function in a five-dimensional random flight problem. We point out a probable connection to Riordan arrays.
用σ n {sigma ＿n}表示{Graham, Knuth和Patashnik的名著《具体数学》意义上的第n个Stirling多项式。我们证明了存在x σ n (x) =∑j = 0 n (2 j j !)−1q n−j (j)x j x}{sigma ＿n}{}left (x)= {sum ＿j}=0{^}n {}{left ({2}^{jj}!)^}-{1q＿n}{-}j {}left (j)x^{j}与j次{多项式}qj {q＿j}。{j.}由此推导出多项式恒等式∑a + b + c + d = n(−1)d (x−2 a−2 b) 3n−s−a−c a !B !C !D !(3n−s−a−c) !=0，对于s∈Z≥1,sum ＿a+b+c+d=n {}{left (-1)^d }{}frac{{left(x-2a-2b)}^{3n-s-a-c}}{a!b!c!d!left(3n-s-a-c)!} =0, hspace{1.0em}hspace{0.1em}text{for}hspace{0.1em}hspace{0.33em} s in{{mathbb{Z}}} ＿ {ge 1，是在}试图为一个五维随机飞行问题的密度函数找到一个更简单的公式时发现的。我们指出了与赖尔登数组的可能联系。
{"title":"Representing the Stirling polynomials σn(x) in dependence of n and an application to polynomial zero identities","authors":"Alexander Kovačec, Pedro Barata de Tovar Sá","doi":"10.1515/spma-2022-0184","DOIUrl":"https://doi.org/10.1515/spma-2022-0184","url":null,"abstract":"Abstract Denote by σ n {sigma }_{n} the n-th Stirling polynomial in the sense of the well-known book Concrete Mathematics by Graham, Knuth and Patashnik. We show that there exist developments x σ n ( x ) = ∑ j = 0 n ( 2 j j ! ) − 1 q n − j ( j ) x j x{sigma }_{n}left(x)={sum }_{j=0}^{n}{left({2}^{j}j!)}^{-1}{q}_{n-j}left(j){x}^{j} with polynomials q j {q}_{j} of degree j . j. We deduce from this the polynomial identities ∑ a + b + c + d = n ( − 1 ) d ( x − 2 a − 2 b ) 3 n − s − a − c a ! b ! c ! d ! ( 3 n − s − a − c ) ! = 0 , for s ∈ Z ≥ 1 , sum _{a+b+c+d=n}{left(-1)}^{d}frac{{left(x-2a-2b)}^{3n-s-a-c}}{a!b!c!d!left(3n-s-a-c)!}=0,hspace{1.0em}hspace{0.1em}text{for}hspace{0.1em}hspace{0.33em}sin {{mathbb{Z}}}_{ge 1}, found in an attempt to find a simpler formula for the density function in a five-dimensional random flight problem. We point out a probable connection to Riordan arrays.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48472000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Recently, some Young-type inequalities have been promoted. The purpose of this article is to give further refinements and reverses to them with Kantorovich constants. Simultaneously, according to the scalar result, we have obtained some corresponding operator inequalities and matrix versions, including Hilbert-Schmidt norm, unitary invariant norm, and trace norm can be regarded as Scalar inequality.
{"title":"New versions of refinements and reverses of Young-type inequalities with the Kantorovich constant","authors":"Mohammad H. M. Rashid, F. Bani-Ahmad","doi":"10.1515/spma-2022-0180","DOIUrl":"https://doi.org/10.1515/spma-2022-0180","url":null,"abstract":"Abstract Recently, some Young-type inequalities have been promoted. The purpose of this article is to give further refinements and reverses to them with Kantorovich constants. Simultaneously, according to the scalar result, we have obtained some corresponding operator inequalities and matrix versions, including Hilbert-Schmidt norm, unitary invariant norm, and trace norm can be regarded as Scalar inequality.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48716408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rafael Bru, Maria T. Gassó, Isabel Giménez, Máximo Santana, José Scott
Abstract Let C(A)=A∘A−T {mathcal{C}}left(A)=Acirc {A}^{-T} be the combined matrix of an invertible matrix A A , where ∘ circ means the Hadamard product of matrices. In this work, we study the combined matrix of a nonsingular matrix, which is an H H -matrix whose comparison matrix is singular. In particular, we focus on C(A) {mathcal{C}}left(A) when A A is diagonally equipotent, and we study whether C(A) {mathcal{C}}left(A) is an H H -matrix and to which class it belongs. Moreover, we give some properties on the diagonal dominance of these matrices and on their comparison matrices.
{"title":"Combined matrix of diagonally equipotent matrices","authors":"Rafael Bru, Maria T. Gassó, Isabel Giménez, Máximo Santana, José Scott","doi":"10.1515/spma-2023-0101","DOIUrl":"https://doi.org/10.1515/spma-2023-0101","url":null,"abstract":"Abstract Let <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>A</m:mi> <m:mrow> <m:mo>∘</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>T</m:mi> </m:mrow> </m:msup> </m:math> {mathcal{C}}left(A)=Acirc {A}^{-T} be the combined matrix of an invertible matrix <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>A</m:mi> </m:math> A , where <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>∘</m:mo> </m:mrow> </m:math> circ means the Hadamard product of matrices. In this work, we study the combined matrix of a nonsingular matrix, which is an <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>H</m:mi> </m:math> H -matrix whose comparison matrix is singular. In particular, we focus on <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {mathcal{C}}left(A) when <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>A</m:mi> </m:math> A is diagonally equipotent, and we study whether <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {mathcal{C}}left(A) is an <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>H</m:mi> </m:math> H -matrix and to which class it belongs. Moreover, we give some properties on the diagonal dominance of these matrices and on their comparison matrices.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135400868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A list Λ = { λ 1 , λ 2 , … , λ n } Lambda =left{{lambda }_{1},{lambda }_{2},ldots ,{lambda }_{n}right} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix and is said to be universally realizable (UR), if it is realizable for each possible Jordan canonical form allowed by Λ Lambda . In 1981, Minc proved that if Λ Lambda is diagonalizably positively realizable, then Λ Lambda is UR [Proc. Amer. Math. Society 83 (1981), 665–669]. The question whether this result holds for nonnegative realizations was open for almost 40 years. Recently, two extensions of Mins’s result have been obtained by Soto et al. [Spec. Matrices 6 (2018), 301–309], [Linear Algebra Appl. 587 (2020), 302–313]. In this work, we exploit these extensions to generate new universal realizability criteria. Moreover, we also prove that under certain conditions, the union of two lists UR is also UR, and for certain criteria, if Λ Lambda is UR, then for t ≥ 0 tge 0 , Λ t = { λ 1 + t , λ 2 ± t , λ 3 , … , λ n } {Lambda }_{t}=left{{lambda }_{1}+t,{lambda }_{2}pm t,{lambda }_{3},ldots ,{lambda }_{n}right} is also UR.
{"title":"On new universal realizability criteria","authors":"Luis Arrieta, R. Soto","doi":"10.1515/spma-2022-0177","DOIUrl":"https://doi.org/10.1515/spma-2022-0177","url":null,"abstract":"Abstract A list Λ = { λ 1 , λ 2 , … , λ n } Lambda =left{{lambda }_{1},{lambda }_{2},ldots ,{lambda }_{n}right} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix and is said to be universally realizable (UR), if it is realizable for each possible Jordan canonical form allowed by Λ Lambda . In 1981, Minc proved that if Λ Lambda is diagonalizably positively realizable, then Λ Lambda is UR [Proc. Amer. Math. Society 83 (1981), 665–669]. The question whether this result holds for nonnegative realizations was open for almost 40 years. Recently, two extensions of Mins’s result have been obtained by Soto et al. [Spec. Matrices 6 (2018), 301–309], [Linear Algebra Appl. 587 (2020), 302–313]. In this work, we exploit these extensions to generate new universal realizability criteria. Moreover, we also prove that under certain conditions, the union of two lists UR is also UR, and for certain criteria, if Λ Lambda is UR, then for t ≥ 0 tge 0 , Λ t = { λ 1 + t , λ 2 ± t , λ 3 , … , λ n } {Lambda }_{t}=left{{lambda }_{1}+t,{lambda }_{2}pm t,{lambda }_{3},ldots ,{lambda }_{n}right} is also UR.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45116834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let M n {{bf{M}}}_{n} be the set of all n × n ntimes n real matrices. A nonsingular matrix A ∈ M n Ain {{bf{M}}}_{n} is called a G-matrix if there exist nonsingular diagonal matrices D 1 {D}_{1} and D 2 {D}_{2} such that A − T = D 1 A D 2 {A}^{-T}={D}_{1}A{D}_{2} . For fixed nonsingular diagonal matrices D 1 {D}_{1} and D 2 {D}_{2} , let G ( D 1 , D 2 ) = { A ∈ M n : A − T = D 1 A D 2 } , {mathbb{G}}left({D}_{1},{D}_{2})=left{Ain {{bf{M}}}_{n}:{A}^{-T}={D}_{1}A{D}_{2}right}, which is called a G-class. The purpose of this short article is to answer the following open question in the affirmative: do there exist two n × n ntimes n G-classes having finite intersection when n ≥ 3 nge 3 ?
抽象让M n{{男朋友{}}}{n套》成为了所有的n×n n 时报matrices庄园。A nonsingular矩阵A M∈n 在{{{M}}} {n}的男朋友打电话是G-matrix如果有存在的对角线nonsingular matrices D D{1}{1}的D和D 2{}{2}如此那A−T = D 1 A D 2 {A} ^ {T} = {D}{1}的A的D{}{2}。为固定的对角线nonsingular matrices D D{1}{1}的D和D 2{}{2},让G D 1, D (2) = {A M∈n: A−T = 1 D A D 2}, {G mathbb{}} 向左拐的D({}{1},{}{2}的)= D左派在{{{A n的男朋友{M}}} {}: {A} ^ {T} = {D}{1}的A的D{}{2}对),这是叫A G-class。《这个短文章的目的是为了回答跟踪开放《affirmative: do有问题存在两个n×n n 时报G-classes玩得有限的intersection当n≥3 ge 3 ?
{"title":"Two n × n G-classes of matrices having finite intersection","authors":"Setareh Golshan, A. Armandnejad, Frank J. Hall","doi":"10.1515/spma-2022-0178","DOIUrl":"https://doi.org/10.1515/spma-2022-0178","url":null,"abstract":"Abstract Let M n {{bf{M}}}_{n} be the set of all n × n ntimes n real matrices. A nonsingular matrix A ∈ M n Ain {{bf{M}}}_{n} is called a G-matrix if there exist nonsingular diagonal matrices D 1 {D}_{1} and D 2 {D}_{2} such that A − T = D 1 A D 2 {A}^{-T}={D}_{1}A{D}_{2} . For fixed nonsingular diagonal matrices D 1 {D}_{1} and D 2 {D}_{2} , let G ( D 1 , D 2 ) = { A ∈ M n : A − T = D 1 A D 2 } , {mathbb{G}}left({D}_{1},{D}_{2})=left{Ain {{bf{M}}}_{n}:{A}^{-T}={D}_{1}A{D}_{2}right}, which is called a G-class. The purpose of this short article is to answer the following open question in the affirmative: do there exist two n × n ntimes n G-classes having finite intersection when n ≥ 3 nge 3 ?","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43276777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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