Abstract In this article, we express the eigenvalues of real antitridiagonal Hankel matrices as the zeros of given rational functions. We still derive eigenvectors for these structured matrices at the expense of prescribed eigenvalues.
{"title":"On the spectral properties of real antitridiagonal Hankel matrices","authors":"J. Lita da Silva","doi":"10.1515/spma-2022-0174","DOIUrl":"https://doi.org/10.1515/spma-2022-0174","url":null,"abstract":"Abstract In this article, we express the eigenvalues of real antitridiagonal Hankel matrices as the zeros of given rational functions. We still derive eigenvectors for these structured matrices at the expense of prescribed eigenvalues.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"64 2","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41290481","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 Monotone matrices are stochastic matrices that satisfy the monotonicity conditions as introduced by Daley in 1968. Monotone Markov chains are useful in modeling phenomena in several areas. Most previous work examines the embedding problem for Markov chains within the entire set of stochastic transition matrices, and only a few studies focus on the embeddability within a specific subset of stochastic matrices. This article examines the embedding in a discrete-time monotone Markov chain, i.e., the existence of monotone matrix roots. Monotone matrix roots of ( 2 × 2 ) left(2times 2) monotone matrices are investigated in previous work. For ( 3 × 3 ) left(3times 3) monotone matrices, this article proves properties that are useful in studying the existence of monotone roots. Furthermore, we demonstrate that all ( 3 × 3 ) left(3times 3) monotone matrices with positive eigenvalues have an m m th root that satisfies the monotonicity conditions (for all values m ∈ N , m ≥ 2 min {mathbb{N}},mge 2 ). For monotone matrices of order n > 3 ngt 3 , diverse scenarios regarding the matrix roots are pointed out, and interesting properties are discussed for block diagonal and diagonalizable monotone matrices.
{"title":"On monotone Markov chains and properties of monotone matrix roots","authors":"M. Guerry","doi":"10.1515/spma-2022-0172","DOIUrl":"https://doi.org/10.1515/spma-2022-0172","url":null,"abstract":"Abstract Monotone matrices are stochastic matrices that satisfy the monotonicity conditions as introduced by Daley in 1968. Monotone Markov chains are useful in modeling phenomena in several areas. Most previous work examines the embedding problem for Markov chains within the entire set of stochastic transition matrices, and only a few studies focus on the embeddability within a specific subset of stochastic matrices. This article examines the embedding in a discrete-time monotone Markov chain, i.e., the existence of monotone matrix roots. Monotone matrix roots of ( 2 × 2 ) left(2times 2) monotone matrices are investigated in previous work. For ( 3 × 3 ) left(3times 3) monotone matrices, this article proves properties that are useful in studying the existence of monotone roots. Furthermore, we demonstrate that all ( 3 × 3 ) left(3times 3) monotone matrices with positive eigenvalues have an m m th root that satisfies the monotonicity conditions (for all values m ∈ N , m ≥ 2 min {mathbb{N}},mge 2 ). For monotone matrices of order n > 3 ngt 3 , diverse scenarios regarding the matrix roots are pointed out, and interesting properties are discussed for block diagonal and diagonalizable monotone matrices.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48474428","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 In this paper, we derive some relationships between the determinants of some special lower Hessenberg matrices whose entries are the terms of certain sequences and the generating functions of these sequences. Moreover, our results are generalizations of the earlier results from previous researches. Furthermore, interesting examples of the determinants of some special lower Hessenberg matrices are presented.
{"title":"Determinants of some Hessenberg matrices with generating functions","authors":"U. Leerawat, K. Daowsud","doi":"10.1515/spma-2022-0170","DOIUrl":"https://doi.org/10.1515/spma-2022-0170","url":null,"abstract":"Abstract In this paper, we derive some relationships between the determinants of some special lower Hessenberg matrices whose entries are the terms of certain sequences and the generating functions of these sequences. Moreover, our results are generalizations of the earlier results from previous researches. Furthermore, interesting examples of the determinants of some special lower Hessenberg matrices are presented.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"11 1","pages":"1 - 8"},"PeriodicalIF":0.5,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45727328","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 In the theory of line graphs of undirected graphs, there exists an important theorem linking the incidence matrix of the root graph to the adjacency matrix of its line graph. For directed or mixed graphs, however, there exists no analogous result. The goal of this article is to present aligned definitions of the adjacency matrix, the incidence matrix, and line graph of a mixed graph such that the mentioned theorem is valid for mixed graphs.
{"title":"Incidence matrices and line graphs of mixed graphs","authors":"Mohammad Abudayah, O. Alomari, T. Sander","doi":"10.1515/spma-2022-0176","DOIUrl":"https://doi.org/10.1515/spma-2022-0176","url":null,"abstract":"Abstract In the theory of line graphs of undirected graphs, there exists an important theorem linking the incidence matrix of the root graph to the adjacency matrix of its line graph. For directed or mixed graphs, however, there exists no analogous result. The goal of this article is to present aligned definitions of the adjacency matrix, the incidence matrix, and line graph of a mixed graph such that the mentioned theorem is valid for mixed graphs.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48202952","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 In this article, we investigate connected signed graphs which have a connected star complement for both −2-2 and 2 (i.e. simultaneously for the two eigenvalues), where −2-2 (resp. 2) is the least (largest) eigenvalue of the adjacency matrix of a signed graph under consideration. We determine all such star complements and their maximal extensions (again, relative to both eigenvalues). As an application, we provide a new proof of the result which identifies all signed graphs that have no eigenvalues other than −2-2 and 2.
{"title":"Star complements for ±2 in signed graphs","authors":"R. Mulas, Z. Stanić","doi":"10.1515/spma-2022-0161","DOIUrl":"https://doi.org/10.1515/spma-2022-0161","url":null,"abstract":"Abstract In this article, we investigate connected signed graphs which have a connected star complement for both −2-2 and 2 (i.e. simultaneously for the two eigenvalues), where −2-2 (resp. 2) is the least (largest) eigenvalue of the adjacency matrix of a signed graph under consideration. We determine all such star complements and their maximal extensions (again, relative to both eigenvalues). As an application, we provide a new proof of the result which identifies all signed graphs that have no eigenvalues other than −2-2 and 2.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"258 - 266"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46679433","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 We study the infinite Jacobi block matrices under the discrete Miura-type transformations which relate matrix Volterra and Toda lattice systems to each other and the situations when the deficiency indices of the corresponding operators are the same. A special attention is paid to the completely indeterminate case (i.e., then the deficiency indices of the corresponding block Jacobi operators are maximal). It is shown that there exists a Miura transformation which retains the complete indeterminacy of Jacobi block matrices appearing in the Lax representation for such systems, namely, if the Lax matrix of Volterra system is completely indeterminate, then so is the Lax matrix of the corresponding Toda system, and vice versa. We consider an implication of the obtained results to the study of matrix orthogonal polynomials as well as to the analysis of self-adjointness of scalar Jacobi operators.
{"title":"Deficiency indices of block Jacobi matrices and Miura transformation","authors":"A. Osipov","doi":"10.1515/spma-2022-0160","DOIUrl":"https://doi.org/10.1515/spma-2022-0160","url":null,"abstract":"Abstract We study the infinite Jacobi block matrices under the discrete Miura-type transformations which relate matrix Volterra and Toda lattice systems to each other and the situations when the deficiency indices of the corresponding operators are the same. A special attention is paid to the completely indeterminate case (i.e., then the deficiency indices of the corresponding block Jacobi operators are maximal). It is shown that there exists a Miura transformation which retains the complete indeterminacy of Jacobi block matrices appearing in the Lax representation for such systems, namely, if the Lax matrix of Volterra system is completely indeterminate, then so is the Lax matrix of the corresponding Toda system, and vice versa. We consider an implication of the obtained results to the study of matrix orthogonal polynomials as well as to the analysis of self-adjointness of scalar Jacobi operators.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"234 - 250"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42372336","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 Flexible systems are obtained from systems of linear equations by adding to the elements of the coefficient matrix and the right-hand side scalar neutrices, which are convex groups of (non-standard) real numbers. The neutrices model imprecisions, giving rise to calculation rules extending informal error calculus. Stability conditions for flexible systems are given in terms of relative imprecision and size of determinants. We then apply the explicit formula for the elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure to find the solution of flexible systems, keeping track of the error terms at every stage. The solution respects the original imprecisions in the right-hand side and is the same as the one given by Cramer’s rule.
{"title":"The explicit formula for Gauss-Jordan elimination applied to flexible systems","authors":"N. Tran, Júlia Justino, I. Berg","doi":"10.1515/spma-2022-0168","DOIUrl":"https://doi.org/10.1515/spma-2022-0168","url":null,"abstract":"Abstract Flexible systems are obtained from systems of linear equations by adding to the elements of the coefficient matrix and the right-hand side scalar neutrices, which are convex groups of (non-standard) real numbers. The neutrices model imprecisions, giving rise to calculation rules extending informal error calculus. Stability conditions for flexible systems are given in terms of relative imprecision and size of determinants. We then apply the explicit formula for the elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure to find the solution of flexible systems, keeping track of the error terms at every stage. The solution respects the original imprecisions in the right-hand side and is the same as the one given by Cramer’s rule.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"366 - 393"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46900813","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 DD be a digraph with vertex set VV and arc set EE. For a vertex uu, the out-degree and in-degree of uu are denoted by du+{d}_{u}^{+} and du−{d}_{u}^{-}, respectively. A vertex-degree-based (VDB) topological index φvarphi is defined for DD as φ(D)=12∑uv∈Eφdu+,dv−,varphi (D)=frac{1}{2}sum _{uvin E}{varphi }_{{d}_{u}^{+},{d}_{v}^{-}}, where φi,j{varphi }_{i,j} is an appropriate function which satisfies φi,j=φj,i{varphi }_{i,j}={varphi }_{j,i}. In this work, we introduce the energy ℰφ(D){{mathcal{ {mathcal E} }}}_{varphi }(D) of a digraph DD with respect to a general VDB topological index φvarphi , and after comparing it with the energy of the underlying graph of its splitting digraph, we derive upper and lower bounds for ℰφ{{mathcal{ {mathcal E} }}}_{varphi } and characterize the digraphs which attain these bounds.
{"title":"Energy of a digraph with respect to a VDB topological index","authors":"Juan Monsalve, J. Rada","doi":"10.1515/spma-2022-0171","DOIUrl":"https://doi.org/10.1515/spma-2022-0171","url":null,"abstract":"Abstract Let DD be a digraph with vertex set VV and arc set EE. For a vertex uu, the out-degree and in-degree of uu are denoted by du+{d}_{u}^{+} and du−{d}_{u}^{-}, respectively. A vertex-degree-based (VDB) topological index φvarphi is defined for DD as φ(D)=12∑uv∈Eφdu+,dv−,varphi (D)=frac{1}{2}sum _{uvin E}{varphi }_{{d}_{u}^{+},{d}_{v}^{-}}, where φi,j{varphi }_{i,j} is an appropriate function which satisfies φi,j=φj,i{varphi }_{i,j}={varphi }_{j,i}. In this work, we introduce the energy ℰφ(D){{mathcal{ {mathcal E} }}}_{varphi }(D) of a digraph DD with respect to a general VDB topological index φvarphi , and after comparing it with the energy of the underlying graph of its splitting digraph, we derive upper and lower bounds for ℰφ{{mathcal{ {mathcal E} }}}_{varphi } and characterize the digraphs which attain these bounds.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"417 - 426"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47544238","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 In this paper, we study the interplay between the structural and spectral properties of the comaximal graph Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) of the ring Zn{{mathbb{Z}}}_{n} for n>2ngt 2. We first determine the structure of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) and deduce some of its properties. We then use the structure of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) to deduce the Laplacian eigenvalues of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) for various nn. We show that Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) is Laplacian integral for n=pαqβn={p}^{alpha }{q}^{beta }, where p,qp,q are primes and α,βalpha ,beta are non-negative integers and hence calculate the number of spanning trees of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) for n=pαqβn={p}^{alpha }{q}^{beta }. The algebraic and vertex connectivity of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) have been shown to be equal for all nn. An upper bound on the second largest Laplacian eigenvalue of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) has been obtained, and a necessary and sufficient condition for its equality has also been determined. Finally, we discuss the multiplicity of the Laplacian spectral radius and the multiplicity of the algebraic connectivity of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}). We then investigate some properties and vertex connectivity of an induced subgraph of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}). Some problems have been discussed at the end of this paper for further research.
{"title":"Laplacian spectrum of comaximal graph of the ring ℤn","authors":"Subarsha Banerjee","doi":"10.1515/spma-2022-0163","DOIUrl":"https://doi.org/10.1515/spma-2022-0163","url":null,"abstract":"Abstract In this paper, we study the interplay between the structural and spectral properties of the comaximal graph Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) of the ring Zn{{mathbb{Z}}}_{n} for n>2ngt 2. We first determine the structure of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) and deduce some of its properties. We then use the structure of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) to deduce the Laplacian eigenvalues of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) for various nn. We show that Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) is Laplacian integral for n=pαqβn={p}^{alpha }{q}^{beta }, where p,qp,q are primes and α,βalpha ,beta are non-negative integers and hence calculate the number of spanning trees of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) for n=pαqβn={p}^{alpha }{q}^{beta }. The algebraic and vertex connectivity of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) have been shown to be equal for all nn. An upper bound on the second largest Laplacian eigenvalue of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}) has been obtained, and a necessary and sufficient condition for its equality has also been determined. Finally, we discuss the multiplicity of the Laplacian spectral radius and the multiplicity of the algebraic connectivity of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}). We then investigate some properties and vertex connectivity of an induced subgraph of Γ(Zn)Gamma left({{mathbb{Z}}}_{n}). Some problems have been discussed at the end of this paper for further research.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"285 - 298"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47307087","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 In this article, we determine upper and lower bounds for the spectral radius of nonnegative matrices. Introducing the notion of average 4-row sum of a nonnegative matrix, we extend various existing formulas for spectral radius bounds. We also refer to their equality cases if the matrix is irreducible, and we present numerical examples to make comparisons among them. Finally, we provide an application to special matrices such as the generalized Fibonacci matrices, which are widely used in applied mathematics and computer science problems.
{"title":"Bounds for the spectral radius of nonnegative matrices and generalized Fibonacci matrices","authors":"Maria Adam, Aikaterini Aretaki","doi":"10.1515/spma-2022-0165","DOIUrl":"https://doi.org/10.1515/spma-2022-0165","url":null,"abstract":"Abstract In this article, we determine upper and lower bounds for the spectral radius of nonnegative matrices. Introducing the notion of average 4-row sum of a nonnegative matrix, we extend various existing formulas for spectral radius bounds. We also refer to their equality cases if the matrix is irreducible, and we present numerical examples to make comparisons among them. Finally, we provide an application to special matrices such as the generalized Fibonacci matrices, which are widely used in applied mathematics and computer science problems.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"308 - 326"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46754449","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：