Pub Date : 2021-01-01DOI: 10.1080/03081087.2022.2049186
Wenxuan Ding, Matthew Ingwersen, Charles R. Johnson
Abstract In the study of eigenvalues, multiplicities, and graphs, the minimum number of multiplicities equal to 1 in a real symmetric matrix with graph G, U(G), is an important constraint on the possible multiplicity lists among matrices in 𝒮(G). Of course, the structure of G must determine U(G), but, even for trees, this linkage has proven elusive. If T is a tree, U(T) is at least 2, but may be much greater. For linear trees, recent work has improved our understanding. Here, we consider nonlinear trees, segregated by diameter. This leads to a new combinatorial construct called a core, for which we are able to calculate U(T). We suspect this bounds U(T) for all nonlinear trees with the given core. In the process, we develop considerable combinatorial information about cores.
{"title":"The minimum number of multiplicity 1 eigenvalues among real symmetric matrices whose graph is a nonlinear tree","authors":"Wenxuan Ding, Matthew Ingwersen, Charles R. Johnson","doi":"10.1080/03081087.2022.2049186","DOIUrl":"https://doi.org/10.1080/03081087.2022.2049186","url":null,"abstract":"Abstract In the study of eigenvalues, multiplicities, and graphs, the minimum number of multiplicities equal to 1 in a real symmetric matrix with graph G, U(G), is an important constraint on the possible multiplicity lists among matrices in 𝒮(G). Of course, the structure of G must determine U(G), but, even for trees, this linkage has proven elusive. If T is a tree, U(T) is at least 2, but may be much greater. For linear trees, recent work has improved our understanding. Here, we consider nonlinear trees, segregated by diameter. This leads to a new combinatorial construct called a core, for which we are able to calculate U(T). We suspect this bounds U(T) for all nonlinear trees with the given core. In the process, we develop considerable combinatorial information about cores.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"203 - 218"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48501159","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, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0,∑k=0nBkk!H(n.k,α)=αH(n+1,1,α)-H(n,1,α),sumlimits_{k = 0}^n {{{{B_k}} over {k!}}Hleft( {n.k,alpha } right) = alpha Hleft( {n + 1,1,alpha } right) - Hleft( {n,1,alpha } right)} ,and for n > r ≥ 0, ∑k=rn-1(-1)ks(k,r)r!αkk!Hn-k(α)=(-1)rH(n,r,α),sumlimits_{k = r}^{n - 1} {{{left( { - 1} right)}^k}{{sleft( {k,r} right)r!} over {{alpha ^k}k!}}{H_{n - k}}left( alpha right) = {{left( { - 1} right)}^r}Hleft( {n,r,alpha } right)} , where Bernoulli numbers Bn and Stirling numbers of the first kind s (n, r).
{"title":"On identities involving generalized harmonic, hyperharmonic and special numbers with Riordan arrays","authors":"S. Koparal, N. Ömür, Ö. Duran","doi":"10.1515/spma-2020-0111","DOIUrl":"https://doi.org/10.1515/spma-2020-0111","url":null,"abstract":"Abstract In this paper, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0,∑k=0nBkk!H(n.k,α)=αH(n+1,1,α)-H(n,1,α),sumlimits_{k = 0}^n {{{{B_k}} over {k!}}Hleft( {n.k,alpha } right) = alpha Hleft( {n + 1,1,alpha } right) - Hleft( {n,1,alpha } right)} ,and for n > r ≥ 0, ∑k=rn-1(-1)ks(k,r)r!αkk!Hn-k(α)=(-1)rH(n,r,α),sumlimits_{k = r}^{n - 1} {{{left( { - 1} right)}^k}{{sleft( {k,r} right)r!} over {{alpha ^k}k!}}{H_{n - k}}left( alpha right) = {{left( { - 1} right)}^r}Hleft( {n,r,alpha } right)} , where Bernoulli numbers Bn and Stirling numbers of the first kind s (n, r).","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"22 - 30"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0111","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42336866","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 Continuous-time Markov chains have transition matrices that vary continuously in time. Classical theory of nonnegative matrices, M-matrices and matrix exponentials is used in the literature to study their dynamics, probability distributions and other stochastic properties. For the benefit of Perron-Frobenius cognoscentes, this theory is surveyed and further adapted to study continuous-time Markov chains on finite state spaces.
{"title":"Matrix Analysis for Continuous-Time Markov Chains","authors":"H. Le, M. Tsatsomeros","doi":"10.1515/spma-2021-0157","DOIUrl":"https://doi.org/10.1515/spma-2021-0157","url":null,"abstract":"Abstract Continuous-time Markov chains have transition matrices that vary continuously in time. Classical theory of nonnegative matrices, M-matrices and matrix exponentials is used in the literature to study their dynamics, probability distributions and other stochastic properties. For the benefit of Perron-Frobenius cognoscentes, this theory is surveyed and further adapted to study continuous-time Markov chains on finite state spaces.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"219 - 233"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45774464","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 third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured. A 1991 paper wrongly suggested that his conjecture is a direct consequence of an 1854 result put forth by Sylvester. Here we recount some of the arguments that led Schrödinger to consider this particular matrix and what might have led to the wrong suggestion. We then give a self-contained elementary (though computational) proof which would have been accessible to Schrödinger. It needs only partial fraction decomposition. We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries. It also allows to prove Schrödinger’s conjecture.
在他1926年的著名论文《Quantisierung als Eigenwertproblem》的第三部分中，Schrödinger遇到了他推测了特征值的参数化三对角矩阵族。1991年的一篇论文错误地认为，他的猜想是西尔维斯特1854年提出的一个结果的直接结果。在这里，我们重述了一些导致Schrödinger考虑这个特殊矩阵的论点，以及可能导致错误建议的原因。然后，我们给出了一个自包含的初等证明(尽管是计算性的)，该证明可以访问Schrödinger。它只需要部分分式分解。最后，我们概述了近几十年来建立的Hahn类正交多项式系统与某些分数阶三对角矩阵之间的联系。它还可以证明Schrödinger的猜想。
{"title":"Schrödinger’s tridiagonal matrix","authors":"A. Kovacec","doi":"10.1515/spma-2020-0124","DOIUrl":"https://doi.org/10.1515/spma-2020-0124","url":null,"abstract":"Abstract In the third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured. A 1991 paper wrongly suggested that his conjecture is a direct consequence of an 1854 result put forth by Sylvester. Here we recount some of the arguments that led Schrödinger to consider this particular matrix and what might have led to the wrong suggestion. We then give a self-contained elementary (though computational) proof which would have been accessible to Schrödinger. It needs only partial fraction decomposition. We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries. It also allows to prove Schrödinger’s conjecture.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"149 - 165"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0124","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44470566","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 By using combinatorial techniques, we obtain an extension of the matrix-tree theorem for general symmetric M-matrices with no restrictions, this means that we do not have to assume the diagonally dominance hypothesis. We express the group inverse of a symmetric M–matrix in terms of the weight of spanning rooted forests. In fact, we give a combinatorial expression for both the determinant of the considered matrix and the determinant of any submatrix obtained by deleting a row and a column. Moreover, the singular case is obtained as a limit case when certain parameter goes to zero. In particular, we recover some known results regarding trees. As examples that illustrate our results we give the expressions for the Group inverse of any symmetric M-matrix of order two and three. We also consider the case of the cycle C4 an example of a non-contractible situation topologically different from a tree. Finally, we obtain some relations between combinatorial numbers, such as Horadam, Fibonacci or Pell numbers and the number of spanning rooted trees on a path.
{"title":"A combinatorial expression for the group inverse of symmetric M-matrices","authors":"Á. Carmona, A. Encinas, M. Mitjana","doi":"10.1515/spma-2020-0137","DOIUrl":"https://doi.org/10.1515/spma-2020-0137","url":null,"abstract":"Abstract By using combinatorial techniques, we obtain an extension of the matrix-tree theorem for general symmetric M-matrices with no restrictions, this means that we do not have to assume the diagonally dominance hypothesis. We express the group inverse of a symmetric M–matrix in terms of the weight of spanning rooted forests. In fact, we give a combinatorial expression for both the determinant of the considered matrix and the determinant of any submatrix obtained by deleting a row and a column. Moreover, the singular case is obtained as a limit case when certain parameter goes to zero. In particular, we recover some known results regarding trees. As examples that illustrate our results we give the expressions for the Group inverse of any symmetric M-matrix of order two and three. We also consider the case of the cycle C4 an example of a non-contractible situation topologically different from a tree. Finally, we obtain some relations between combinatorial numbers, such as Horadam, Fibonacci or Pell numbers and the number of spanning rooted trees on a path.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"275 - 296"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0137","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49274718","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}
D. Carter, K.E. DiMarco, C. Johnson, L. Wedemeyer, Z. Yu
Abstract The 3-by-n TP-completable patterns are characterized by identifying the minimal obstructions up to natural symmetries. They are finite in number.
摘要3-by-n TP可完成模式的特征在于识别达到自然对称的最小障碍。它们的数量是有限的。
{"title":"The Totally Positive Completion Problem: The 3-by-n Case","authors":"D. Carter, K.E. DiMarco, C. Johnson, L. Wedemeyer, Z. Yu","doi":"10.1515/spma-2020-0134","DOIUrl":"https://doi.org/10.1515/spma-2020-0134","url":null,"abstract":"Abstract The 3-by-n TP-completable patterns are characterized by identifying the minimal obstructions up to natural symmetries. They are finite in number.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"226 - 239"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0134","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49256356","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 derive an inequality that includes the largest eigenvalue of the adjacency matrix and walks of an arbitrary length of a signed graph. We also consider certain particular cases.
{"title":"A note on a walk-based inequality for the index of a signed graph","authors":"Z. Stanić","doi":"10.1515/spma-2020-0120","DOIUrl":"https://doi.org/10.1515/spma-2020-0120","url":null,"abstract":"Abstract We derive an inequality that includes the largest eigenvalue of the adjacency matrix and walks of an arbitrary length of a signed graph. We also consider certain particular cases.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"19 - 21"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0120","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46908060","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 introduce most of the concepts for q-Lie algebras in a way independent of the base field K. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q-roots for certain well-known q-Lie groups contain an extra q-addition, and consequently, for most of the quantities which are q-deformed, we add a prefix q in the respective name. Important examples are the q-Cartan subalgebra and the q-Cartan Killing form. We introduce the concept q-homogeneous spaces in a formal way exemplified by the examples SUq(1,1)SOq(2){{S{U_q}left( {1,1} right)} over {S{O_q}left( 2 right)}} and SOq(3)SOq(2){{S{O_q}left( 3 right)} over {S{O_q}left( 2 right)}} with corresponding q-Lie groups and q-geodesics. By introducing a q-deformed semidirect product, we can define exact sequences of q-Lie groups and some other interesting q-homogeneous spaces. We give an example of the corresponding q-Iwasawa decomposition for SLq(2).
{"title":"Further results on q-Lie groups, q-Lie algebras and q-homogeneous spaces","authors":"T. Ernst","doi":"10.1515/spma-2020-0129","DOIUrl":"https://doi.org/10.1515/spma-2020-0129","url":null,"abstract":"Abstract We introduce most of the concepts for q-Lie algebras in a way independent of the base field K. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q-roots for certain well-known q-Lie groups contain an extra q-addition, and consequently, for most of the quantities which are q-deformed, we add a prefix q in the respective name. Important examples are the q-Cartan subalgebra and the q-Cartan Killing form. We introduce the concept q-homogeneous spaces in a formal way exemplified by the examples SUq(1,1)SOq(2){{S{U_q}left( {1,1} right)} over {S{O_q}left( 2 right)}} and SOq(3)SOq(2){{S{O_q}left( 3 right)} over {S{O_q}left( 2 right)}} with corresponding q-Lie groups and q-geodesics. By introducing a q-deformed semidirect product, we can define exact sequences of q-Lie groups and some other interesting q-homogeneous spaces. We give an example of the corresponding q-Iwasawa decomposition for SLq(2).","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"119 - 148"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0129","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46601509","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 L be the infinite lower triangular Toeplitz matrix with first column (µ, a1, a2, ..., ap, a1, ..., ap, ...)T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ, a1, a2, are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ, a1, ..., ap. It depends on the asymptotics in µ of the l2-norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.
{"title":"The smallest singular value of certain Toeplitz-related parametric triangular matrices","authors":"M. S. Solary, A. Kovacec, S. Capizzano","doi":"10.1515/spma-2020-0127","DOIUrl":"https://doi.org/10.1515/spma-2020-0127","url":null,"abstract":"Abstract Let L be the infinite lower triangular Toeplitz matrix with first column (µ, a1, a2, ..., ap, a1, ..., ap, ...)T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ, a1, a2, are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ, a1, ..., ap. It depends on the asymptotics in µ of the l2-norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"103 - 111"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0127","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41484511","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 Given a real number a ≥ 1, let Kn(a) be the set of all n × n unit lower triangular matrices with each element in the interval [−a, a]. Denoting by λn(·) the smallest eigenvalue of a given matrix, let cn(a) = min {λ n(YYT) : Y ∈ Kn(a)}. Then cn(a)sqrt {{c_n}left( a right)} is the smallest singular value in Kn(a). We find all minimizing matrices. Moreover, we study the asymptotic behavior of cn(a) as n → ∞. Finally, replacing [−a, a] with [a, b], a ≤ 0 < b, we present an open question: Can our results be generalized in this extension?
{"title":"On the smallest singular value in the class of unit lower triangular matrices with entries in [−a, a]","authors":"E. Altinisik","doi":"10.1515/spma-2020-0139","DOIUrl":"https://doi.org/10.1515/spma-2020-0139","url":null,"abstract":"Abstract Given a real number a ≥ 1, let Kn(a) be the set of all n × n unit lower triangular matrices with each element in the interval [−a, a]. Denoting by λn(·) the smallest eigenvalue of a given matrix, let cn(a) = min {λ n(YYT) : Y ∈ Kn(a)}. Then cn(a)sqrt {{c_n}left( a right)} is the smallest singular value in Kn(a). We find all minimizing matrices. Moreover, we study the asymptotic behavior of cn(a) as n → ∞. Finally, replacing [−a, a] with [a, b], a ≤ 0 < b, we present an open question: Can our results be generalized in this extension?","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"297 - 304"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0139","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43814451","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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