Abstract We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix A A is completely positive. Our decomposition can be applied to a wide range of matrices. We give alternate proofs for a number of related results found in the literature in a simple, straightforward manner. We show that the cp-rank of any completely positive irreducible tridiagonal doubly stochastic matrix is equal to its rank. We then consider symmetric pentadiagonal matrices, proving some analogous results and providing two different decompositions sufficient for complete positivity. We illustrate our constructions with a number of examples.
{"title":"The complete positivity of symmetric tridiagonal and pentadiagonal matrices","authors":"Lei Cao, Darian Mclaren, S. Plosker","doi":"10.1515/spma-2022-0173","DOIUrl":"https://doi.org/10.1515/spma-2022-0173","url":null,"abstract":"Abstract We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix A A is completely positive. Our decomposition can be applied to a wide range of matrices. We give alternate proofs for a number of related results found in the literature in a simple, straightforward manner. We show that the cp-rank of any completely positive irreducible tridiagonal doubly stochastic matrix is equal to its rank. We then consider symmetric pentadiagonal matrices, proving some analogous results and providing two different decompositions sufficient for complete positivity. We illustrate our constructions with a number of examples.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44938240","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 It is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu = λu leads to a matrix eigenvalue problem (EVP) Ax =λx where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs) Ax = λBx which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.
{"title":"Analytical solutions to some generalized and polynomial eigenvalue problems","authors":"Quanling Deng","doi":"10.1515/spma-2020-0135","DOIUrl":"https://doi.org/10.1515/spma-2020-0135","url":null,"abstract":"Abstract It is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu = λu leads to a matrix eigenvalue problem (EVP) Ax =λx where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs) Ax = λBx which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"240 - 256"},"PeriodicalIF":0.5,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0135","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49081840","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 lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size −3, −5, −7, . . . . For such paths, we find the generating functions of them, according to length, ending at level i, both, when considering them from left to right and from right to left. The generating functions are intrinsically cubic, and thus (for i = 0) in bijection to various objects, like even trees, ternary trees, etc.
{"title":"Generating functions for a lattice path model introduced by Deutsch","authors":"H. Prodinger","doi":"10.1515/spma-2020-0133","DOIUrl":"https://doi.org/10.1515/spma-2020-0133","url":null,"abstract":"Abstract The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size −3, −5, −7, . . . . For such paths, we find the generating functions of them, according to length, ending at level i, both, when considering them from left to right and from right to left. The generating functions are intrinsically cubic, and thus (for i = 0) in bijection to various objects, like even trees, ternary trees, etc.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"217 - 225"},"PeriodicalIF":0.5,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0133","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45465800","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 Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and distance matrix. We conclude by enumerating the prevelance of this construction in small graphs for the adjacency matrix, combinatorial Laplacian matrix, and distance Laplacian matrix.
{"title":"Cospectral constructions for several graph matrices using cousin vertices","authors":"Kate J. Lorenzen","doi":"10.1515/spma-2020-0143","DOIUrl":"https://doi.org/10.1515/spma-2020-0143","url":null,"abstract":"Abstract Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and distance matrix. We conclude by enumerating the prevelance of this construction in small graphs for the adjacency matrix, combinatorial Laplacian matrix, and distance Laplacian matrix.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"9 - 22"},"PeriodicalIF":0.5,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0143","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42036388","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 (λ, v) be a known real eigenpair of an n×n real matrix A. In this paper it is shown how to locate the other eigenvalues of A in terms of the components of v. The obtained region is a union of Gershgorin discs of the second type recently introduced by the authors in a previous paper. Two cases are considered depending on whether or not some of the components of v are equal to zero. Upper bounds are obtained, in two different ways, for the largest eigenvalue in absolute value of A other than. Detailed examples are provided. Although nonnegative irreducible matrices are somewhat emphasized, the main results in this paper are valid for any n × n real matrix with n≥3.
{"title":"Inclusion regions and bounds for the eigenvalues of matrices with a known eigenpair","authors":"Rachid Marsli, Frank J. Hall","doi":"10.1515/spma-2020-0115","DOIUrl":"https://doi.org/10.1515/spma-2020-0115","url":null,"abstract":"Abstract Let (λ, v) be a known real eigenpair of an n×n real matrix A. In this paper it is shown how to locate the other eigenvalues of A in terms of the components of v. The obtained region is a union of Gershgorin discs of the second type recently introduced by the authors in a previous paper. Two cases are considered depending on whether or not some of the components of v are equal to zero. Upper bounds are obtained, in two different ways, for the largest eigenvalue in absolute value of A other than. Detailed examples are provided. Although nonnegative irreducible matrices are somewhat emphasized, the main results in this paper are valid for any n × n real matrix with n≥3.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"204 - 220"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0115","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47305836","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 spectrum of two interesting stochastic matrices appearing in an engineering paper is completely determined. As a result, an inequality conjectured in that paper, involving two second largest eigenvalues, is easily proved.
{"title":"The spectrum of two interesting stochastic matrices","authors":"N. Anghel","doi":"10.1515/spma-2020-0003","DOIUrl":"https://doi.org/10.1515/spma-2020-0003","url":null,"abstract":"Abstract The spectrum of two interesting stochastic matrices appearing in an engineering paper is completely determined. As a result, an inequality conjectured in that paper, involving two second largest eigenvalues, is easily proved.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"17 - 21"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44795752","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 f be an operator monotonic function on I and A, B∈I (H), the class of all selfadjoint operators with spectra in I. Assume that p : [0.1], →ℝ is non-decreasing on [0, 1]. In this paper we obtained, among others, that for A ≤ B and f an operator monotonic function on I, 0≤∫01p(t)f((1-t)A+tB)dt-∫01p(t)dt∫01f((1-t)A+tB)dt≤14[ p(1)-p(0) ][ f(B)-f(A) ] matrix{0 hfill & { le intlimits_0^1 {pleft( t right)fleft( {left( {1 - t} right)A + tB} right)dt - intlimits_0^1 {pleft( t right)dtintlimits_0^1 {fleft( {left( {1 - t} right)A + tB} right)dt} } } } hfill cr {} hfill & { le {1 over 4}left[ {pleft( 1 right) - pleft( 0 right)} right]left[ {fleft( B right) - fleft( A right)} right]} hfill cr } in the operator order. Several other similar inequalities for either p or f is differentiable, are also provided. Applications for power function and logarithm are given as well.
摘要设f是I上的算子单调函数，A, B∈I (H)，是I上的所有谱自伴随算子的类，设p:[0.1]，→在[0,1]上不递减。本文得到了对于A≤B和f在I上的算子单调函数，0≤∫01p(t)f((1-t)A+tB)dt-∫01p(t) A+tB)dt≤14[p(1)-p(0)][f(B)-f(A)] matrix{0 hfill & { le intlimits_0^1 {pleft( t right)fleft( {left( {1 - t} right)A + tB} right)dt - intlimits_0^1 {pleft( t right)dtintlimits_0^1 {fleft( {left( {1 - t} right)A + tB} right)dt} } } } hfill cr {} hfill & { le {1 over 4}left[ {pleft( 1 right) - pleft( 0 right)} right]left[ {fleft( B right) - fleft( A right)} right]} hfill cr }的算子阶。对于p或f均可微，也给出了其他几个类似的不等式。并给出了幂函数和对数的应用。
{"title":"Some integral inequalities for operator monotonic functions on Hilbert spaces","authors":"S. Dragomir","doi":"10.1515/spma-2020-0108","DOIUrl":"https://doi.org/10.1515/spma-2020-0108","url":null,"abstract":"Abstract Let f be an operator monotonic function on I and A, B∈I (H), the class of all selfadjoint operators with spectra in I. Assume that p : [0.1], →ℝ is non-decreasing on [0, 1]. In this paper we obtained, among others, that for A ≤ B and f an operator monotonic function on I, 0≤∫01p(t)f((1-t)A+tB)dt-∫01p(t)dt∫01f((1-t)A+tB)dt≤14[ p(1)-p(0) ][ f(B)-f(A) ] matrix{0 hfill & { le intlimits_0^1 {pleft( t right)fleft( {left( {1 - t} right)A + tB} right)dt - intlimits_0^1 {pleft( t right)dtintlimits_0^1 {fleft( {left( {1 - t} right)A + tB} right)dt} } } } hfill cr {} hfill & { le {1 over 4}left[ {pleft( 1 right) - pleft( 0 right)} right]left[ {fleft( B right) - fleft( A right)} right]} hfill cr } in the operator order. Several other similar inequalities for either p or f is differentiable, are also provided. Applications for power function and logarithm are given as well.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"172 - 180"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0108","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41382054","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 Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field 𝔽q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinants of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted
{"title":"Determinants of some special matrices over commutative finite chain rings","authors":"Somphong Jitman","doi":"10.1515/spma-2020-0118","DOIUrl":"https://doi.org/10.1515/spma-2020-0118","url":null,"abstract":"Abstract Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field 𝔽q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinants of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"242 - 256"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0118","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42375993","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 An important decomposition for unitary matrices, the CMV-decomposition, is extended to general non-unitary matrices. This relates to short recurrence relations constructing biorthogonal bases for a particular pair of extended Krylov subspaces.
{"title":"Non-unitary CMV-decomposition","authors":"Niel Van Buggenhout, M. Van Barel, R. Vandebril","doi":"10.1515/spma-2020-0107","DOIUrl":"https://doi.org/10.1515/spma-2020-0107","url":null,"abstract":"Abstract An important decomposition for unitary matrices, the CMV-decomposition, is extended to general non-unitary matrices. This relates to short recurrence relations constructing biorthogonal bases for a particular pair of extended Krylov subspaces.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"144 - 159"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0107","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47142515","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 It is shown that in any TP matrix, a line (row or column) with two speci˝ed entries in any positions (and the others appropriately chosen) may be inserted in any position, as long as the two entries are consistent with total positivity. This generalizes an unconstrained result previously proven, and the two may not generally be increased to three or more. Applications are given, and this fact should be useful in other completion problems, as the unconstrained result has been.
{"title":"Doubly constrained totally positive line insertion","authors":"Charles R. Johnson, David W. Allen","doi":"10.1515/spma-2020-0112","DOIUrl":"https://doi.org/10.1515/spma-2020-0112","url":null,"abstract":"Abstract It is shown that in any TP matrix, a line (row or column) with two speci˝ed entries in any positions (and the others appropriately chosen) may be inserted in any position, as long as the two entries are consistent with total positivity. This generalizes an unconstrained result previously proven, and the two may not generally be increased to three or more. Applications are given, and this fact should be useful in other completion problems, as the unconstrained result has been.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"181 - 185"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0112","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42375041","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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