Abstract We prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.
{"title":"A trace bound for integer-diagonal positive semidefinite matrices","authors":"Lon H. Mitchell","doi":"10.1515/spma-2020-0002","DOIUrl":"https://doi.org/10.1515/spma-2020-0002","url":null,"abstract":"Abstract We prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"14 - 16"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46907648","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 Three combinatorial matrices were considered and their LU-decompositions were found. This is typically done by (creative) guessing, and the proofs are more or less routine calculations.
{"title":"Some combinatorial matrices and their LU-decomposition","authors":"H. Prodinger","doi":"10.1515/spma-2020-0007","DOIUrl":"https://doi.org/10.1515/spma-2020-0007","url":null,"abstract":"Abstract Three combinatorial matrices were considered and their LU-decompositions were found. This is typically done by (creative) guessing, and the proofs are more or less routine calculations.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"61 - 67"},"PeriodicalIF":0.5,"publicationDate":"2019-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48748698","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 evaluate determinants of “block hook” matrices, which are block matrices consist of hook matrices. In particular, we deduce that the determinant of a block hook matrix factorizes nicely. In addition we give a combinatorial interpretation of the aforesaid factorization property by counting weighted paths in a suitable weighted digraph.
{"title":"Enumeration of weighted paths on a digraph and block hook determinant","authors":"S. Bera","doi":"10.1515/spma-2020-0130","DOIUrl":"https://doi.org/10.1515/spma-2020-0130","url":null,"abstract":"Abstract In this article, we evaluate determinants of “block hook” matrices, which are block matrices consist of hook matrices. In particular, we deduce that the determinant of a block hook matrix factorizes nicely. In addition we give a combinatorial interpretation of the aforesaid factorization property by counting weighted paths in a suitable weighted digraph.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"197 - 211"},"PeriodicalIF":0.5,"publicationDate":"2019-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0130","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48141563","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 define signless Laplacian matrix of a hypergraph and obtain structural properties from its eigenvalues. We generalize several known results for graphs, relating the spectrum of this matrix to structural parameters of the hypergraph such as the maximum degree, diameter, and the chromatic number. In addition, we characterize the complete signless Laplacian spectrum for the class of power hypergraphs from the spectrum of its base hypergraph.
{"title":"The signless Laplacian matrix of hypergraphs","authors":"Kaue Cardoso, V. Trevisan","doi":"10.1515/spma-2022-0166","DOIUrl":"https://doi.org/10.1515/spma-2022-0166","url":null,"abstract":"Abstract In this article, we define signless Laplacian matrix of a hypergraph and obtain structural properties from its eigenvalues. We generalize several known results for graphs, relating the spectrum of this matrix to structural parameters of the hypergraph such as the maximum degree, diameter, and the chromatic number. In addition, we characterize the complete signless Laplacian spectrum for the class of power hypergraphs from the spectrum of its base hypergraph.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"327 - 342"},"PeriodicalIF":0.5,"publicationDate":"2019-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48073207","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 A real symmetric matrix A is said to be completely positive if it can be written as BBt for some (not necessarily square) nonnegative matrix B. A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidefinite is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. We compute a matrix such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and . This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of .
{"title":"Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse","authors":"Joyentanuj Das, Sachindranath Jayaraman, Sumit Mohanty","doi":"10.1515/spma-2020-0109","DOIUrl":"https://doi.org/10.1515/spma-2020-0109","url":null,"abstract":"Abstract A real symmetric matrix A is said to be completely positive if it can be written as BBt for some (not necessarily square) nonnegative matrix B. A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidefinite is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. We compute a matrix such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and . This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of .","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"160 - 171"},"PeriodicalIF":0.5,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0109","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46764179","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}
Linh T. Duong, Brenda Kroschel, Michael Riddell, K. V. Vander Meulen, Adam Van Tuyl
Abstract The zero forcing number of a graph has been applied to communication complexity, electrical power grid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of various circulant graphs, including families of bipartite circulants, as well as all cubic circulants. We extend the definition of the Möbius ladder to a type of torus product to obtain bounds on the minimum rank and the maximum nullity on these products. We obtain equality for torus products by employing orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined these numbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds for all circulant graphs.
{"title":"Maximum nullity and zero forcing of circulant graphs","authors":"Linh T. Duong, Brenda Kroschel, Michael Riddell, K. V. Vander Meulen, Adam Van Tuyl","doi":"10.1515/spma-2020-0106","DOIUrl":"https://doi.org/10.1515/spma-2020-0106","url":null,"abstract":"Abstract The zero forcing number of a graph has been applied to communication complexity, electrical power grid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of various circulant graphs, including families of bipartite circulants, as well as all cubic circulants. We extend the definition of the Möbius ladder to a type of torus product to obtain bounds on the minimum rank and the maximum nullity on these products. We obtain equality for torus products by employing orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined these numbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds for all circulant graphs.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"221 - 234"},"PeriodicalIF":0.5,"publicationDate":"2019-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0106","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49627433","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 singular values σ > 1 of an n × n involutory matrix A appear in pairs (σ, 1σ {1 over sigma } ). Their left and right singular vectors are closely connected. The case of singular values σ = 1 is discussed in detail. These singular values may appear in pairs (1,1) with closely connected left and right singular vectors or by themselves. The link between the left and right singular vectors is used to reformulate the singular value decomposition (SVD) of an involutory matrix as an eigendecomposition. This displays an interesting relation between the singular values of an involutory matrix and its eigenvalues. Similar observations hold for the SVD, the singular values and the coneigenvalues of (skew-)coninvolutory matrices.
{"title":"On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices","authors":"H. Faßbender, Martin Halwass","doi":"10.1515/spma-2020-0001","DOIUrl":"https://doi.org/10.1515/spma-2020-0001","url":null,"abstract":"Abstract The singular values σ > 1 of an n × n involutory matrix A appear in pairs (σ, 1σ {1 over sigma } ). Their left and right singular vectors are closely connected. The case of singular values σ = 1 is discussed in detail. These singular values may appear in pairs (1,1) with closely connected left and right singular vectors or by themselves. The link between the left and right singular vectors is used to reformulate the singular value decomposition (SVD) of an involutory matrix as an eigendecomposition. This displays an interesting relation between the singular values of an involutory matrix and its eigenvalues. Similar observations hold for the SVD, the singular values and the coneigenvalues of (skew-)coninvolutory matrices.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"8 1","pages":"1 - 13"},"PeriodicalIF":0.5,"publicationDate":"2019-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49343380","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 Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank 1 idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) U is a symmetric unitary matrix if and only if it has the form I − 2E for a symmetric idempotent E, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.
{"title":"Unique builders for classes of matrices","authors":"T. Hurley","doi":"10.1515/spma-2020-0122","DOIUrl":"https://doi.org/10.1515/spma-2020-0122","url":null,"abstract":"Abstract Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank 1 idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) U is a symmetric unitary matrix if and only if it has the form I − 2E for a symmetric idempotent E, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"52 - 65"},"PeriodicalIF":0.5,"publicationDate":"2019-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0122","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45772853","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 distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, (G) = I − D(G)−1/2A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.
{"title":"The normalized distance Laplacian","authors":"Carolyn Reinhart","doi":"10.1515/spma-2020-0114","DOIUrl":"https://doi.org/10.1515/spma-2020-0114","url":null,"abstract":"Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, (G) = I − D(G)−1/2A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"1 - 18"},"PeriodicalIF":0.5,"publicationDate":"2019-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0114","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45781111","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 Generalized matrix functions were first introduced in [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163-171]. Recently, it has been recognized that these matrix functions arise in a number of applications, and various numerical methods have been proposed for their computation. The exploitation of structural properties, when present, can lead to more efficient and accurate algorithms. The main goal of this paper is to identify structural properties of matrices which are preserved by generalized matrix functions. In cases where a given property is not preserved in general, we provide conditions on the underlying scalar function under which the property of interest will be preserved by the corresponding generalized matrix function.
{"title":"Some matrix properties preserved by generalized matrix functions","authors":"M. Benzi, Ru Huang","doi":"10.1515/SPMA-2019-0003","DOIUrl":"https://doi.org/10.1515/SPMA-2019-0003","url":null,"abstract":"Abstract Generalized matrix functions were first introduced in [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163-171]. Recently, it has been recognized that these matrix functions arise in a number of applications, and various numerical methods have been proposed for their computation. The exploitation of structural properties, when present, can lead to more efficient and accurate algorithms. The main goal of this paper is to identify structural properties of matrices which are preserved by generalized matrix functions. In cases where a given property is not preserved in general, we provide conditions on the underlying scalar function under which the property of interest will be preserved by the corresponding generalized matrix function.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"7 1","pages":"27 - 37"},"PeriodicalIF":0.5,"publicationDate":"2019-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/SPMA-2019-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43794424","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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