A. Abiad, Boris Brimkov, Jane Breen, T. R. Cameron, H. Gupta, R. R. Villagr'an
Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper, we show that several NP-hard zero forcing numbers are not characterized by the spectra of several types of associated matrices with a graph. In particular, we consider standard zero forcing, positive semidefinite zero forcing, and skew zero forcing and provide constructions of infinite families of pairs of cospectral graphs, which have different values for these numbers. We explore several methods for obtaining these cospectral graphs including using graph products, graph joins, and graph switching. Among these, we provide a construction involving regular adjacency cospectral graphs; the regularity of this construction also implies cospectrality with respect to several other matrices including the Laplacian, signless Laplacian, and normalized Laplacian. We also provide a construction where pairs of cospectral graphs can have an arbitrarily large difference between their zero forcing numbers.
{"title":"Constructions of cospectral graphs with different zero forcing numbers","authors":"A. Abiad, Boris Brimkov, Jane Breen, T. R. Cameron, H. Gupta, R. R. Villagr'an","doi":"10.13001/ela.2022.6737","DOIUrl":"https://doi.org/10.13001/ela.2022.6737","url":null,"abstract":"Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper, we show that several NP-hard zero forcing numbers are not characterized by the spectra of several types of associated matrices with a graph. In particular, we consider standard zero forcing, positive semidefinite zero forcing, and skew zero forcing and provide constructions of infinite families of pairs of cospectral graphs, which have different values for these numbers. We explore several methods for obtaining these cospectral graphs including using graph products, graph joins, and graph switching. Among these, we provide a construction involving regular adjacency cospectral graphs; the regularity of this construction also implies cospectrality with respect to several other matrices including the Laplacian, signless Laplacian, and normalized Laplacian. We also provide a construction where pairs of cospectral graphs can have an arbitrarily large difference between their zero forcing numbers.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43360105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper is devoted to the study of the maximal angle between the $5times 5$ semidefinite matrix cone and $5times 5$ nonnegative matrix cone. A signomial geometric programming problem is formulated in the process to find the maximal angle. Instead of using an optimization problem solver to solve the problem numerically, the method of Lagrange Multipliers is used to solve the signomial geometric program, and therefore, to find the maximal angle between these two cones.
{"title":"The maximal angle between 5×5 positive semidefinite and 5×5 nonnegative matrices","authors":"Qinghong Zhang","doi":"10.13001/ela.2021.6647","DOIUrl":"https://doi.org/10.13001/ela.2021.6647","url":null,"abstract":"The paper is devoted to the study of the maximal angle between the $5times 5$ semidefinite matrix cone and $5times 5$ nonnegative matrix cone. A signomial geometric programming problem is formulated in the process to find the maximal angle. Instead of using an optimization problem solver to solve the problem numerically, the method of Lagrange Multipliers is used to solve the signomial geometric program, and therefore, to find the maximal angle between these two cones.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43223600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $m$ and $n$ be positive integers, and let $R =(r_1, ldots, r_m)$ and $S =(s_1,ldots, s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m times n$ $(0,1)$-matrix where for each $i$, the $i^{th}$ row of $F(R,S)$ consists of $r_i$ 1's followed by $n-r_i$ 0's. Let $Ain A(R,S)$. The discrepancy of A, $disc(A)$, is the number of positions in which $F(R)$ has a 1 and $A$ has a 0. In this paper, we investigate the possible discrepancy of $A^t$ versus the discrepancy of $A$. We show that if the discrepancy of $A$ is $ell$, then the discrepancy of the transpose of $A$ is at least $frac{ell}{2}$ and at most $2ell$. These bounds are tight.
{"title":"(0,1)-matrices and Discrepancy","authors":"LeRoy B. Beasley","doi":"10.13001/ela.2021.5033","DOIUrl":"https://doi.org/10.13001/ela.2021.5033","url":null,"abstract":" Let $m$ and $n$ be positive integers, and let $R =(r_1, ldots, r_m)$ and $S =(s_1,ldots, s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m times n$ $(0,1)$-matrix where for each $i$, the $i^{th}$ row of $F(R,S)$ consists of $r_i$ 1's followed by $n-r_i$ 0's. Let $Ain A(R,S)$. The discrepancy of A, $disc(A)$, is the number of positions in which $F(R)$ has a 1 and $A$ has a 0. In this paper, we investigate the possible discrepancy of $A^t$ versus the discrepancy of $A$. We show that if the discrepancy of $A$ is $ell$, then the discrepancy of the transpose of $A$ is at least $frac{ell}{2}$ and at most $2ell$. These bounds are tight.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46537470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Bueno, S. Furtado, Aelita Klausmeier, Joey Veltri
In this paper, a complete description of the linear maps $phi:W_{n}rightarrow W_{n}$ that preserve the Lorentz spectrum is given when $n=2$, and $W_{n}$ is the space $M_{n}$ of $ntimes n$ real matrices or the subspace $S_{n}$ of $M_{n}$ formed by the symmetric matrices. In both cases, it has been shown that $phi(A)=PAP^{-1}$ for all $Ain W_{2}$, where $P$ is a matrix with a certain structure. It was also shown that such preservers do not change the nature of the Lorentz eigenvalues (that is, the fact that they are associated with Lorentz eigenvectors in the interior or on the boundary of the Lorentz cone). These results extend to $n=2$ those for $ngeq 3$ obtained by Bueno, Furtado, and Sivakumar (2021). The case $n=2$ has some specificities, when compared to the case $ngeq3,$ due to the fact that the Lorentz cone in $mathbb{R}^{2}$ is polyedral, contrary to what happens when it is contained in $mathbb{R}^{n}$ with $ngeq3.$ Thus, the study of the Lorentz spectrum preservers on $W_n = M_n$ also follows from the known description of the Pareto spectrum preservers on $M_n$.
{"title":"Linear maps preserving the Lorentz spectrum: the $2 times 2$ case","authors":"M. Bueno, S. Furtado, Aelita Klausmeier, Joey Veltri","doi":"10.13001/ela.2022.6925","DOIUrl":"https://doi.org/10.13001/ela.2022.6925","url":null,"abstract":"In this paper, a complete description of the linear maps $phi:W_{n}rightarrow W_{n}$ that preserve the Lorentz spectrum is given when $n=2$, and $W_{n}$ is the space $M_{n}$ of $ntimes n$ real matrices or the subspace $S_{n}$ of $M_{n}$ formed by the symmetric matrices. In both cases, it has been shown that $phi(A)=PAP^{-1}$ for all $Ain W_{2}$, where $P$ is a matrix with a certain structure. It was also shown that such preservers do not change the nature of the Lorentz eigenvalues (that is, the fact that they are associated with Lorentz eigenvectors in the interior or on the boundary of the Lorentz cone). These results extend to $n=2$ those for $ngeq 3$ obtained by Bueno, Furtado, and Sivakumar (2021). The case $n=2$ has some specificities, when compared to the case $ngeq3,$ due to the fact that the Lorentz cone in $mathbb{R}^{2}$ is polyedral, contrary to what happens when it is contained in $mathbb{R}^{n}$ with $ngeq3.$ Thus, the study of the Lorentz spectrum preservers on $W_n = M_n$ also follows from the known description of the Pareto spectrum preservers on $M_n$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48920811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A list $Lambda ={lambda_{1},ldots,lambda_{n}}$ of complex numbers is said to be realizable, if it is the spectrum of an entrywise nonnegative matrix $A$. In this case, $A$ is said to be a realizing matrix. $Lambda$ is said to be universally realizable, if it is realizable for each possible Jordan canonical form (JCF) allowed by $Lambda$. The problem of the universal realizability of spectra is called the universal realizability problem (URP). Here, we study the centrosymmetric URP, that is, the problem of finding a nonnegative centrosymmetric matrix for each JCF allowed by a given list $Lambda $. In particular, sufficient conditions for the centrosymmetric URP to have a solution are generated.
{"title":"Centrosymmetric universal realizability","authors":"Ana Julio, Yankis R. Linares, R. Soto","doi":"10.13001/ela.2021.5781","DOIUrl":"https://doi.org/10.13001/ela.2021.5781","url":null,"abstract":"A list $Lambda ={lambda_{1},ldots,lambda_{n}}$ of complex numbers is said to be realizable, if it is the spectrum of an entrywise nonnegative matrix $A$. In this case, $A$ is said to be a realizing matrix. $Lambda$ is said to be universally realizable, if it is realizable for each possible Jordan canonical form (JCF) allowed by $Lambda$. The problem of the universal realizability of spectra is called the universal realizability problem (URP). Here, we study the centrosymmetric URP, that is, the problem of finding a nonnegative centrosymmetric matrix for each JCF allowed by a given list $Lambda $. In particular, sufficient conditions for the centrosymmetric URP to have a solution are generated.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42680422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explore algebraic and spectral properties of weighted graphs containing twin vertices that are useful in quantum state transfer. We extend the notion of adjacency strong cospectrality to Hermitian matrices, with focus on the generalized adjacency matrix and the generalized normalized adjacency matrix. We then determine necessary and sufficient conditions such that a pair of twin vertices in a weighted graph exhibits strong cospectrality with respect to the above-mentioned matrices. We also determine when strong cospectrality is preserved under Cartesian and direct products of graphs. Moreover, we generalize known results about equitable and almost equitable partitions and use these to determine which joins of the form $Xvee H$, where $X$ is either the complete or empty graph, exhibit strong cospectrality.
{"title":"Strong cospectrality and twin vertices in weighted graphs","authors":"Hermie Monterde","doi":"10.13001/ela.2022.6721","DOIUrl":"https://doi.org/10.13001/ela.2022.6721","url":null,"abstract":"We explore algebraic and spectral properties of weighted graphs containing twin vertices that are useful in quantum state transfer. We extend the notion of adjacency strong cospectrality to Hermitian matrices, with focus on the generalized adjacency matrix and the generalized normalized adjacency matrix. We then determine necessary and sufficient conditions such that a pair of twin vertices in a weighted graph exhibits strong cospectrality with respect to the above-mentioned matrices. We also determine when strong cospectrality is preserved under Cartesian and direct products of graphs. Moreover, we generalize known results about equitable and almost equitable partitions and use these to determine which joins of the form $Xvee H$, where $X$ is either the complete or empty graph, exhibit strong cospectrality.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45828404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we focus the attention on the Hadamard factorization problem for Hurwitz polynomials. We give a new necessary condition for Hadamard factorizability of Hurwitz stable polynomials of degree $ngeq 4$ and show that for $n= 4$ this condition is also sufficient. The effectiveness of the result is illustrated during construction of examples of stable polynomials that are not Hadamard factorizable.
{"title":"Simple necessary conditions for Hadamard factorizability of Hurwitz polynomials","authors":"S. Bialas, M. Góra","doi":"10.13001/ela.2021.5957","DOIUrl":"https://doi.org/10.13001/ela.2021.5957","url":null,"abstract":"In this paper, we focus the attention on the Hadamard factorization problem for Hurwitz polynomials. We give a new necessary condition for Hadamard factorizability of Hurwitz stable polynomials of degree $ngeq 4$ and show that for $n= 4$ this condition is also sufficient. The effectiveness of the result is illustrated during construction of examples of stable polynomials that are not Hadamard factorizable.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42045718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Dockter, Pietro Paparella, R. L. Perry, Jonathan D Ta
An invertible matrix is called a Perron similarity if one of its columns and the corresponding row of its inverse are both nonnegative or both nonpositive. Such matrices are of relevance and import in the study of the nonnegative inverse eigenvalue problem. In this work, Kronecker products of Perron similarities are examined and used to construct ideal Perron similarities all of whose rows are extremal.
{"title":"Kronecker products of Perron similarities","authors":"J. Dockter, Pietro Paparella, R. L. Perry, Jonathan D Ta","doi":"10.13001/ela.2022.6697","DOIUrl":"https://doi.org/10.13001/ela.2022.6697","url":null,"abstract":"An invertible matrix is called a Perron similarity if one of its columns and the corresponding row of its inverse are both nonnegative or both nonpositive. Such matrices are of relevance and import in the study of the nonnegative inverse eigenvalue problem. In this work, Kronecker products of Perron similarities are examined and used to construct ideal Perron similarities all of whose rows are extremal.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41800747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
McDonald and Paparella [Linear Algebra Appl. 498 (2016), 145-159] gave a necessary condition on the structure of the Jordan chains of h-cyclic matrices. In this work, that necessary condition is shown to be sufficient. As a consequence, we provide a spectral characterization of nonsingular, h-cyclic matrices.
{"title":"Jordan chains of h-cyclic matrices, II","authors":"Andrew L. Nickerson, Pietro Paparella","doi":"10.13001/ela.2022.7019","DOIUrl":"https://doi.org/10.13001/ela.2022.7019","url":null,"abstract":"McDonald and Paparella [Linear Algebra Appl. 498 (2016), 145-159] gave a necessary condition on the structure of the Jordan chains of h-cyclic matrices. In this work, that necessary condition is shown to be sufficient. As a consequence, we provide a spectral characterization of nonsingular, h-cyclic matrices.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42673377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work studies the kernel of a linear operator associated with the generalized k-fold commutator. Given a set $mathfrak{A}= left{ A_{1}, ldots ,A_{k} right}$ of real $n times n$ matrices, the commutator is denoted by$[A_{1}| ldots |A_{k}]$. For a fixed set of matrices $mathfrak{A}$ we introduce a multilinear skew-symmetric linear operator $T_{mathfrak{A}}(X)=T(A_{1}, ldots ,A_{k})[X]=[A_{1}| ldots |A_{k} |X] $. For fixed $n$ and $k ge 2n-1, ; T_{mathfrak{A}} equiv 0$ by the Amitsur--Levitski Theorem [2] , which motivated this work. The matrix representation $M$ of the linear transformation $T$ is called the k-commutator matrix. $M$ has interesting properties, e.g., it is a commutator; for $k$ odd, there is a permutation of the rows of $M$ that makes it skew-symmetric. For both $k$ and $n$ odd, a provocative matrix $mathcal{S}$ appears in the kernel of $T$. By using the Moore--Penrose inverse and introducing a conjecture about the rank of $M$, the entries of $mathcal{S}$ are shown to be quotients of polynomials in the entries of the matrices in $mathfrak{A}$. One case of the conjecture has been recently proven by Brassil. The Moore--Penrose inverse provides a full rank decomposition of $M$.
{"title":"Generalized Commutators and the Moore-Penrose Inverse","authors":"I. Pressman","doi":"10.13001/ela.2021.4991","DOIUrl":"https://doi.org/10.13001/ela.2021.4991","url":null,"abstract":"This work studies the kernel of a linear operator associated with the generalized k-fold commutator. Given a set $mathfrak{A}= left{ A_{1}, ldots ,A_{k} right}$ of real $n times n$ matrices, the commutator is denoted by$[A_{1}| ldots |A_{k}]$. For a fixed set of matrices $mathfrak{A}$ we introduce a multilinear skew-symmetric linear operator $T_{mathfrak{A}}(X)=T(A_{1}, ldots ,A_{k})[X]=[A_{1}| ldots |A_{k} |X] $. For fixed $n$ and $k ge 2n-1, ; T_{mathfrak{A}} equiv 0$ by the Amitsur--Levitski Theorem [2] , which motivated this work. The matrix representation $M$ of the linear transformation $T$ is called the k-commutator matrix. $M$ has interesting properties, e.g., it is a commutator; for $k$ odd, there is a permutation of the rows of $M$ that makes it skew-symmetric. For both $k$ and $n$ odd, a provocative matrix $mathcal{S}$ appears in the kernel of $T$. By using the Moore--Penrose inverse and introducing a conjecture about the rank of $M$, the entries of $mathcal{S}$ are shown to be quotients of polynomials in the entries of the matrices in $mathfrak{A}$. One case of the conjecture has been recently proven by Brassil. The Moore--Penrose inverse provides a full rank decomposition of $M$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43127380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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