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Bidiagonalization of (k, k + 1)-tridiagonal matrices (k, k + 1)-三对角矩阵的双对角化
IF 0.5
Special Matrices Pub Date : 2019-01-01 DOI: 10.1515/SPMA-2019-0002
S. Takahira, T. Sogabe, T. Usuda
{"title":"Bidiagonalization of (k, k + 1)-tridiagonal matrices","authors":"S. Takahira, T. Sogabe, T. Usuda","doi":"10.1515/SPMA-2019-0002","DOIUrl":"https://doi.org/10.1515/SPMA-2019-0002","url":null,"abstract":"Abstract In this paper,we present the bidiagonalization of n-by-n (k, k+1)-tridiagonal matriceswhen n < 2k. Moreover,we show that the determinant of an n-by-n (k, k+1)-tridiagonal matrix is the product of the diagonal elements and the eigenvalues of the matrix are the diagonal elements. This paper is related to the fast block diagonalization algorithm using the permutation matrix from [T. Sogabe and M. El-Mikkawy, Appl. Math. Comput., 218, (2011), 2740-2743] and [A. Ohashi, T. Sogabe, and T. S. Usuda, Int. J. Pure and App. Math., 106, (2016), 513-523].","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"20 - 26"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/SPMA-2019-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42270845","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}
引用次数: 14
Extensions of the Eneström-Kakeya theorem for matrix polynomials 矩阵多项式Eneström-Kakeya定理的扩展
IF 0.5
Special Matrices Pub Date : 2019-01-01 DOI: 10.1515/spma-2019-0024
A. Melman
{"title":"Extensions of the Eneström-Kakeya theorem for matrix polynomials","authors":"A. Melman","doi":"10.1515/spma-2019-0024","DOIUrl":"https://doi.org/10.1515/spma-2019-0024","url":null,"abstract":"Abstract The classical Eneström-Kakeya theorem establishes explicit upper and lower bounds on the zeros of a polynomial with positive coefficients and has been generalized for positive definite matrix polynomials by several authors. Recently, extensions that improve the (scalar) Eneström-Kakeya theorem were obtained with a transparent and unified approach using just a few tools. Here, the same tools are used to generalize these extensions to positive definite matrix polynomials, while at the same time generalizing the tools themselves. In the process, a framework is developed that can naturally generate additional similar results.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"304 - 315"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2019-0024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42367780","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}
引用次数: 0
Construction of generalized rotations and quasi-orthogonal matrices 广义旋转与拟正交矩阵的构造
IF 0.5
Special Matrices Pub Date : 2019-01-01 DOI: 10.1515/spma-2019-0010
L. Verde-Star
{"title":"Construction of generalized rotations and quasi-orthogonal matrices","authors":"L. Verde-Star","doi":"10.1515/spma-2019-0010","DOIUrl":"https://doi.org/10.1515/spma-2019-0010","url":null,"abstract":"Abstract We propose some methods for the construction of large quasi-orthogonal matrices and generalized rotations that may be used in applications in data communications and image processing. We use certain combinations of constructions by blocks similar to the one used by Sylvester to build Hadamard matrices. The orthogonal designs related with the matrix representations of the complex numbers, the quaternions, and the octonions are used in our construction procedures.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"107 - 113"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2019-0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44304036","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}
引用次数: 3
Explicit determinants, inverses and eigenvalues of four band Toeplitz matrices with perturbed rows 带扰动行四带Toeplitz矩阵的显式行列式、逆矩阵和特征值
IF 0.5
Special Matrices Pub Date : 2019-01-01 DOI: 10.1515/spma-2019-0004
Maoyun Zhang, Xiaoyu Jiang, Zhaolin Jiang
{"title":"Explicit determinants, inverses and eigenvalues of four band Toeplitz matrices with perturbed rows","authors":"Maoyun Zhang, Xiaoyu Jiang, Zhaolin Jiang","doi":"10.1515/spma-2019-0004","DOIUrl":"https://doi.org/10.1515/spma-2019-0004","url":null,"abstract":"Abstract In this paper, four-band Toeplitz matrices and four-band Hankel matrices of type I and type II with perturbed rows are introduced. Explicit determinants, inverses and eigenvalues for these matrices are derived by using a nice inverse formula of block bidiagonal Toeplitz matrices.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"52 - 66"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2019-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45767207","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}
引用次数: 2
Diagonalizable matrices whose graph is a tree: the minimum number of distinct eigenvalues and the feasibility of eigenvalue assignments 图为树的可对角化矩阵:不同特征值的最小数目和特征值赋值的可行性
IF 0.5
Special Matrices Pub Date : 2019-01-01 DOI: 10.1515/spma-2019-0025
Carlos M. Saiago
{"title":"Diagonalizable matrices whose graph is a tree: the minimum number of distinct eigenvalues and the feasibility of eigenvalue assignments","authors":"Carlos M. Saiago","doi":"10.1515/spma-2019-0025","DOIUrl":"https://doi.org/10.1515/spma-2019-0025","url":null,"abstract":"Abstract Considered are combinatorially symmetric matrices, whose graph is a given tree, in view of the fact recent analysis shows that the geometric multiplicity theory for the eigenvalues of such matrices closely parallels that for real symmetric (and complex Hermitian) matrices. In contrast to the real symmetric case, it is shown that (a) the smallest example (13 vertices) of a tree and multiplicity list (3, 3, 3, 1, 1, 1, 1) meeting standard necessary conditions that has no real symmetric realizations does have a diagonalizable realization and for arbitrary prescribed (real and multiple) eigenvalues, and (b) that all trees with diameter < 8 are geometrically di-minimal (i.e., have diagonalizable realizations with as few of distinct eigenvalues as the diameter). This re-raises natural questions about multiplicity lists that proved subtly false in the real symmetric case. What is their status in the geometric multiplicity list case?","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"316 - 326"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2019-0025","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47324532","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}
引用次数: 4
Numerical construction of structured matrices with given eigenvalues 具有给定特征值的结构化矩阵的数值构造
IF 0.5
Special Matrices Pub Date : 2019-01-01 DOI: 10.1515/spma-2019-0020
Brian D. Sutton
{"title":"Numerical construction of structured matrices with given eigenvalues","authors":"Brian D. Sutton","doi":"10.1515/spma-2019-0020","DOIUrl":"https://doi.org/10.1515/spma-2019-0020","url":null,"abstract":"Abstract We consider a structured inverse eigenvalue problem in which the eigenvalues of a real symmetric matrix are specified and selected entries may be constrained to take specific numerical values or to be nonzero. This includes the problem of specifying the graph of the matrix, which is determined by the locations of zero and nonzero entries. In this article, we develop a numerical method for constructing a solution to the structured inverse eigenvalue problem. The problem is recast as a constrained optimization problem over the orthogonal manifold, and a numerical optimization routine seeks its solution.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"263 - 271"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2019-0020","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48978148","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}
引用次数: 0
A note on the eigenvalue free intervals of some classes of signed threshold graphs 关于一类有符号阈值图的特征值自由区间的注记
IF 0.5
Special Matrices Pub Date : 2019-01-01 DOI: 10.1515/spma-2019-0014
M. Anđelić, T. Koledin, Z. Stanić
{"title":"A note on the eigenvalue free intervals of some classes of signed threshold graphs","authors":"M. Anđelić, T. Koledin, Z. Stanić","doi":"10.1515/spma-2019-0014","DOIUrl":"https://doi.org/10.1515/spma-2019-0014","url":null,"abstract":"Abstract We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q(Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ġ ) – xI (x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"218 - 225"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2019-0014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44771917","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}
引用次数: 0
Some Hermite-Hadamard type inequalities for operator convex functions and positive maps 算子凸函数和正映射的一些Hermite-Hadamard型不等式
IF 0.5
Special Matrices Pub Date : 2019-01-01 DOI: 10.1515/SPMA-2019-0005
S. S. Dragomir
{"title":"Some Hermite-Hadamard type inequalities for operator convex functions and positive maps","authors":"S. S. Dragomir","doi":"10.1515/SPMA-2019-0005","DOIUrl":"https://doi.org/10.1515/SPMA-2019-0005","url":null,"abstract":"Abstract In this paper we establish some inequalities of Hermite-Hadamard type for operator convex functions and positive maps. Applications for power function and logarithm are also provided.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"38 - 51"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/SPMA-2019-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45571997","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}
引用次数: 20
Achievable multiplicity partitions in the inverse eigenvalue problem of a graph 图的特征值反问题中可实现的多重划分
IF 0.5
Special Matrices Pub Date : 2019-01-01 DOI: 10.1515/spma-2019-0022
Mohammad Adm, Shaun M. Fallat, Karen Meagher, S. Nasserasr, S. Plosker, Boting Yang
{"title":"Achievable multiplicity partitions in the inverse eigenvalue problem of a graph","authors":"Mohammad Adm, Shaun M. Fallat, Karen Meagher, S. Nasserasr, S. Plosker, Boting Yang","doi":"10.1515/spma-2019-0022","DOIUrl":"https://doi.org/10.1515/spma-2019-0022","url":null,"abstract":"Abstract Associated to a graph G is a set 𝒮(G) of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If G has n vertices, then the multiplicities of the eigenvalues of any matrix in 𝒮 (G) partition n; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs G for which there is a matrix in 𝒮 (G) with partitions [n − 2, 2] have been characterized. We find families of graphs G for which there is a matrix in 𝒮 (G) with multiplicity partition [n − k, k] for k ≥ 2. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in 𝒮 (G) with multiplicity partition [n − k, k] to show the complexities of characterizing these graphs.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"276 - 290"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2019-0022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49270876","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}
引用次数: 7
Inertias of Laplacian matrices of weighted signed graphs 加权有符号图的拉普拉斯矩阵的不等式
IF 0.5
Special Matrices Pub Date : 2019-01-01 DOI: 10.1515/spma-2019-0026
K. H. Monfared, G. MacGillivray, D. Olesky, P. van den Driessche
{"title":"Inertias of Laplacian matrices of weighted signed graphs","authors":"K. H. Monfared, G. MacGillivray, D. Olesky, P. van den Driessche","doi":"10.1515/spma-2019-0026","DOIUrl":"https://doi.org/10.1515/spma-2019-0026","url":null,"abstract":"Abstract We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs. First we characterize signed graphs with a unique Laplacian inertia. Then we show that there is a sufficiently small perturbation of the nonzero weights on the edges of any connected weighted signed graph so that all eigenvalues of its Laplacian matrix are simple. Next, we give upper bounds on the number of possible Laplacian inertias for signed graphs with a fixed flexibility τ (a combinatorial parameter of signed graphs), and show that these bounds are sharp for an infinite family of signed graphs. Finally, we provide upper bounds for the number of possible Laplacian inertias of signed graphs in terms of the number of vertices.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"327 - 342"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2019-0026","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43037005","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}
引用次数: 0
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