Special MatricesPub Date : 2020-01-01DOI: 10.1515/spma-2020-0103
T. He, P. Shiue
{"title":"A note on Eulerian numbers and Toeplitz matrices","authors":"T. He, P. Shiue","doi":"10.1515/spma-2020-0103","DOIUrl":"https://doi.org/10.1515/spma-2020-0103","url":null,"abstract":"Abstract This note presents a new formula of Eulerian numbers derived from Toeplitz matrices via Riordan array approach.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"123 - 130"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0103","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41862168","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}
Special MatricesPub Date : 2020-01-01DOI: 10.1515/spma-2020-0006
B. Bakhadly, A. Guterman, M. J. de la Puente
{"title":"Orthogonality for (0, −1) tropical normal matrices","authors":"B. Bakhadly, A. Guterman, M. J. de la Puente","doi":"10.1515/spma-2020-0006","DOIUrl":"https://doi.org/10.1515/spma-2020-0006","url":null,"abstract":"Abstract We study pairs of mutually orthogonal normal matrices with respect to tropical multiplication. Minimal orthogonal pairs are characterized. The diameter and girth of three graphs arising from the orthogonality equivalence relation are computed.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"40 - 60"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41364974","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}
Special MatricesPub Date : 2020-01-01DOI: 10.1515/spma-2020-0004
M. Shakil, M. Ahsanullah
{"title":"Some Characterizations of the Distribution of the Condition Number of a Complex Gaussian Matrix","authors":"M. Shakil, M. Ahsanullah","doi":"10.1515/spma-2020-0004","DOIUrl":"https://doi.org/10.1515/spma-2020-0004","url":null,"abstract":"Abstract The objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"22 - 35"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48599812","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}
Special MatricesPub Date : 2020-01-01DOI: 10.1515/spma-2020-0009
Doaa Al-Saafin, J. Garloff
{"title":"Sufficient conditions for symmetric matrices to have exactly one positive eigenvalue","authors":"Doaa Al-Saafin, J. Garloff","doi":"10.1515/spma-2020-0009","DOIUrl":"https://doi.org/10.1515/spma-2020-0009","url":null,"abstract":"Abstract Let A = [aij] be a real symmetric matrix. If f : (0, ∞) → [0, ∞) is a Bernstein function, a sufficient condition for the matrix [f (aij)] to have only one positive eigenvalue is presented. By using this result, new results for a symmetric matrix with exactly one positive eigenvalue, e.g., properties of its Hadamard powers, are derived.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"98 - 103"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45628891","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}
Special MatricesPub Date : 2020-01-01DOI: 10.1515/spma-2020-0012
Yaru Fu, Xiaoyu Jiang, Zhaolin Jiang, S. Jhang
{"title":"Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns","authors":"Yaru Fu, Xiaoyu Jiang, Zhaolin Jiang, S. Jhang","doi":"10.1515/spma-2020-0012","DOIUrl":"https://doi.org/10.1515/spma-2020-0012","url":null,"abstract":"Abstract In this paper, our main attention is paid to calculate the determinants and inverses of two types Toeplitz and Hankel tridiagonal matrices with perturbed columns. Specifically, the determinants of the n × n Toeplitz tridiagonal matrices with perturbed columns (type I, II) can be expressed by using the famous Fibonacci numbers, the inverses of Toeplitz tridiagonal matrices with perturbed columns can also be expressed by using the well-known Lucas numbers and four entries in matrix 𝔸. And the determinants of the n×n Hankel tridiagonal matrices with perturbed columns (type I, II) are (−1]) (-1)n(n-1)2 {left( { - 1} right)^{{{nleft( {n - 1} right)} over 2}}} times of the determinant of the Toeplitz tridiagonal matrix with perturbed columns type I, the entries of the inverses of the Hankel tridiagonal matrices with perturbed columns (type I, II) are the same as that of the inverse of Toeplitz tridiagonal matrix with perturbed columns type I, except the position. In addition, we present some algorithms based on the main theoretical results. Comparison of our new algorithms and some recent works is given. The numerical result confirms our new theoretical results. And we show the superiority of our method by comparing the CPU time of some existing algorithms studied recently.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"131 - 143"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48265201","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}
Special MatricesPub Date : 2020-01-01DOI: 10.1515/spma-2020-0116
Hsin-Yun Ching, Rigoberto Fl'orez, Antara Mukherjee
{"title":"Families of Integral Cographs within a Triangular Array","authors":"Hsin-Yun Ching, Rigoberto Fl'orez, Antara Mukherjee","doi":"10.1515/spma-2020-0116","DOIUrl":"https://doi.org/10.1515/spma-2020-0116","url":null,"abstract":"Abstract The determinant Hosoya triangle, is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle mod 2 gives rise to three infinite families of graphs, that are formed by complete product (join) of (the union of) two complete graphs with an empty graph. We give a necessary and sufficient condition for a graph from these families to be integral. Some features of these graphs are: they are integral cographs, all graphs have at most five distinct eigenvalues, all graphs are either d-regular graphs with d =2, 4, 6, . . . or almost-regular graphs, and some of them are Laplacian integral. Finally we extend some of these results to the Hosoya triangle.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"257 - 273"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0116","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42633208","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}
Special MatricesPub Date : 2020-01-01DOI: 10.1515/spma-2020-0010
I. Fazekas, Sándor Pecsora
{"title":"On the spectrum of noisy blown-up matrices","authors":"I. Fazekas, Sándor Pecsora","doi":"10.1515/spma-2020-0010","DOIUrl":"https://doi.org/10.1515/spma-2020-0010","url":null,"abstract":"Abstract We study the eigenvalues of large perturbed matrices. We consider a pattern matrix P, we blow it up to get a large block-matrix Bn. We can observe only a noisy version of matrix Bn. So we add a random noise Wn to obtain the perturbed matrix An = Bn + Wn. Our aim is to find the structural eigenvalues of An. We prove asymptotic theorems on this problem and also suggest a graphical method to distinguish the structural and the non-structural eigenvalues of An.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"104 - 122"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45969701","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}
Special MatricesPub Date : 2020-01-01DOI: 10.1515/spma-2020-0117
Mohammad Adm, Shaun M. Fallat, Karen Meagher, S. Nasserasr, S. Plosker, Boting Yang
{"title":"Corrigendum to “Achievable Multiplicity partitions in the Inverse Eigenvalue Problem of a graph” [Spec. Matrices 2019; 7:276-290.]","authors":"Mohammad Adm, Shaun M. Fallat, Karen Meagher, S. Nasserasr, S. Plosker, Boting Yang","doi":"10.1515/spma-2020-0117","DOIUrl":"https://doi.org/10.1515/spma-2020-0117","url":null,"abstract":"Abstract We correct an error in the original Lemma 3.4 in our paper “Achievable Multiplicity partitions in the IEVP of a graph”’ [Spec. Matrices 2019; 7:276-290.]. We have re-written Section 3 accordingly.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"235 - 241"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0117","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46577443","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}
Special MatricesPub Date : 2020-01-01DOI: 10.1515/spma-2020-0113
J. McDonald, R. Nandi, K. Sivakumar, P. Sushmitha, M. Tsatsomeros, E. Wendler, Megan Wendler
{"title":"M-matrix and inverse M-matrix extensions","authors":"J. McDonald, R. Nandi, K. Sivakumar, P. Sushmitha, M. Tsatsomeros, E. Wendler, Megan Wendler","doi":"10.1515/spma-2020-0113","DOIUrl":"https://doi.org/10.1515/spma-2020-0113","url":null,"abstract":"Abstract A class of matrices that simultaneously generalizes the M-matrices and the inverse M-matrices is brought forward and its properties are reviewed. It is interesting to see how this class bridges the properties of the matrices it generalizes and provides a new perspective on their classical theory.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"186 - 203"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0113","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48256932","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}
Special MatricesPub Date : 2020-01-01DOI: 10.1515/spma-2020-0005
Lei Cao, Ariana Hall, S. Koyuncu
{"title":"A short note on extreme points of certain polytopes","authors":"Lei Cao, Ariana Hall, S. Koyuncu","doi":"10.1515/spma-2020-0005","DOIUrl":"https://doi.org/10.1515/spma-2020-0005","url":null,"abstract":"Abstract We give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"36 - 39"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47386441","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}