We prove a Frobenius norm inequality for three matrices, analogous to the well-known Bottcher--Wenzel inequality. The situation is also similar: standard inequalities would yield an upper bound, which however can be reduced by means of further, detailed investigations.
{"title":"A norm inequality for three matrices","authors":"L. László","doi":"10.13001/ela.2022.6563","DOIUrl":"https://doi.org/10.13001/ela.2022.6563","url":null,"abstract":"We prove a Frobenius norm inequality for three matrices, analogous to the well-known Bottcher--Wenzel inequality. The situation is also similar: standard inequalities would yield an upper bound, which however can be reduced by means of further, detailed investigations.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45390390","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}
Polynomial matrices $A(lambda)$ and $B(lambda)$ of size $ntimes n$ over a field $mathbb {F}$ are semiscalar equivalent if there exist a nonsingular $ntimes n$ matrix $P$ over $mathbb F$ and an invertible $ntimes n$ matrix $Q(lambda)$ over $mathbb F[lambda]$ such that $A(lambda)=PB(lambda)Q(lambda)$. The aim of this article is to present necessary and sufficient conditions for the semiscalar equivalence of nonsingular matrices $A(lambda)$ and $ B(lambda) $ over a field ${mathbb F }$ of characteristic zero in terms of solutions of a homogenous system of linear equations.
{"title":"A note on semiscalar equivalence of polynomial matrices","authors":"V. Prokip","doi":"10.13001/ela.2022.6505","DOIUrl":"https://doi.org/10.13001/ela.2022.6505","url":null,"abstract":"Polynomial matrices $A(lambda)$ and $B(lambda)$ of size $ntimes n$ over a field $mathbb {F}$ are semiscalar equivalent if there exist a nonsingular $ntimes n$ matrix $P$ over $mathbb F$ and an invertible $ntimes n$ matrix $Q(lambda)$ over $mathbb F[lambda]$ such that $A(lambda)=PB(lambda)Q(lambda)$. The aim of this article is to present necessary and sufficient conditions for the semiscalar equivalence of nonsingular matrices $A(lambda)$ and $ B(lambda) $ over a field ${mathbb F }$ of characteristic zero in terms of solutions of a homogenous system of linear equations.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47629365","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 problems of characterizing sign pattern matrices that allow or require diagonalizability are mostly open. In this paper, we introduce the concept of essential index for a tree sign pattern matrix and use it to investigate the allow problem on diagonalizability for sign pattern matrices having their graphs as trees. We characterize sign pattern matrices allowing diagonalizability, whose graphs are star or path. We also give a sufficient condition for sign pattern matrices whose graphs are trees to allow diagonalizability. Further, we give a necessary condition for a sign pattern matrix to require diagonalizability and characterize all star sign pattern matrices that require diagonalizability.
{"title":"Sign patterns associated with some graphs that allow or require diagonalizability","authors":"Sunil Das","doi":"10.13001/ela.2022.5557","DOIUrl":"https://doi.org/10.13001/ela.2022.5557","url":null,"abstract":"The problems of characterizing sign pattern matrices that allow or require diagonalizability are mostly open. In this paper, we introduce the concept of essential index for a tree sign pattern matrix and use it to investigate the allow problem on diagonalizability for sign pattern matrices having their graphs as trees. We characterize sign pattern matrices allowing diagonalizability, whose graphs are star or path. We also give a sufficient condition for sign pattern matrices whose graphs are trees to allow diagonalizability. Further, we give a necessary condition for a sign pattern matrix to require diagonalizability and characterize all star sign pattern matrices that require diagonalizability.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43613592","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 show a simple method for constructing larger dimension nonnegative matrices with somewhat arbitrary entries which can be irreducible or reducible but preserving the spectral radius via row sum expansions. This yields a sufficient criteria for two square nonnegative matrices of arbitrary dimension to have the same spectral radius, a way to compare spectral radii of two arbitrary square nonnegative matrices, and a way to derive new upper and lower bounds on the spectral radius which give the standard row sum bounds as a special case.
{"title":"Bounds via spectral radius-preserving row sum expansions","authors":"Joseph P. Stover","doi":"10.13001/ela.2022.6981","DOIUrl":"https://doi.org/10.13001/ela.2022.6981","url":null,"abstract":"We show a simple method for constructing larger dimension nonnegative matrices with somewhat arbitrary entries which can be irreducible or reducible but preserving the spectral radius via row sum expansions. This yields a sufficient criteria for two square nonnegative matrices of arbitrary dimension to have the same spectral radius, a way to compare spectral radii of two arbitrary square nonnegative matrices, and a way to derive new upper and lower bounds on the spectral radius which give the standard row sum bounds as a special case.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47389634","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}
Tran Nam Son, Truong Huu Dung, Nguyen Thi Thai Ha, Mai Hoang Bien
Let $F$ be a field and let $n$ be a natural number greater than $1$. The aim of this paper is to prove that if $F$ contains at least three elements, then every matrix in the special linear group $mathrm{SL}_n(F)$ is a product of at most two commutators of involutions.
{"title":"On decompositions of matrices into products of commutators of involutions","authors":"Tran Nam Son, Truong Huu Dung, Nguyen Thi Thai Ha, Mai Hoang Bien","doi":"10.13001/ela.2022.6797","DOIUrl":"https://doi.org/10.13001/ela.2022.6797","url":null,"abstract":"Let $F$ be a field and let $n$ be a natural number greater than $1$. The aim of this paper is to prove that if $F$ contains at least three elements, then every matrix in the special linear group $mathrm{SL}_n(F)$ is a product of at most two commutators of involutions.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49142301","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. Boussaïri, A. Chaïchaâ, B. Chergui, Sara Ezzahir, S. Lakhlifi, Soukaïna Mahzoum
The Slater index $i(T)$ of a tournament $T$ is the minimum number of arcs that must be reversed to make $T$ transitive. In this paper, we define a parameter $Lambda(T)$ from the spectrum of the skew-adjacency matrix of $T$, called the spectral Slater index. This parameter is a measure of remoteness between the spectrum of $T$ and that of a transitive tournament. We show that $Lambda(T)leq8, i(T)$ and we characterize the tournaments with maximal spectral Slater index. As an application, an improved lower bound on the Slater index of doubly regular tournaments is given.
{"title":"Spectral Slater index of tournaments","authors":"A. Boussaïri, A. Chaïchaâ, B. Chergui, Sara Ezzahir, S. Lakhlifi, Soukaïna Mahzoum","doi":"10.13001/ela.2022.6407","DOIUrl":"https://doi.org/10.13001/ela.2022.6407","url":null,"abstract":"The Slater index $i(T)$ of a tournament $T$ is the minimum number of arcs that must be reversed to make $T$ transitive. In this paper, we define a parameter $Lambda(T)$ from the spectrum of the skew-adjacency matrix of $T$, called the spectral Slater index. This parameter is a measure of remoteness between the spectrum of $T$ and that of a transitive tournament. We show that $Lambda(T)leq8, i(T)$ and we characterize the tournaments with maximal spectral Slater index. As an application, an improved lower bound on the Slater index of doubly regular tournaments is given.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47558124","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}
Convex functions have been well studied in the literature for scalars and matrices. However, other types of convex functions have not received the same attention given to the usual convex functions. The main goal of this article is to present matrix inequalities for many types of convex functions, including log-convex, harmonically convex, geometrically convex, and others. The results extend many known results in the literature in this direction. For example, it is shown that if $A,B$ are positive definite matrices and $f$ is a continuous $sigmatau$-convex function on an interval containing the spectra of $A,B$, thenbegin{align*}lambda^downarrow (f(Asigma B))prec_wlambda^downarrow left(f(A)tau f(B)right),end{align*}for the matrix means $sigma,tauin{nabla_{alpha},!_{alpha}}$ and $alphain[0,1]$. Further, if $sigma=sharp_{alpha}$, thenbegin{align*} lambda^downarrow left(fleft(e^{Anabla_{alpha}B}right)right)prec_wlambda^downarrow left(f(e^A)tau f(e^B))right).end{align*}Similar inequalities will be presented for two-variable functions too.
{"title":"Majorization inequalities via convex functions","authors":"M. Kian, M. Sababheh","doi":"10.13001/ela.2022.6901","DOIUrl":"https://doi.org/10.13001/ela.2022.6901","url":null,"abstract":"Convex functions have been well studied in the literature for scalars and matrices. However, other types of convex functions have not received the same attention given to the usual convex functions. The main goal of this article is to present matrix inequalities for many types of convex functions, including log-convex, harmonically convex, geometrically convex, and others. The results extend many known results in the literature in this direction. For example, it is shown that if $A,B$ are positive definite matrices and $f$ is a continuous $sigmatau$-convex function on an interval containing the spectra of $A,B$, thenbegin{align*}lambda^downarrow (f(Asigma B))prec_wlambda^downarrow left(f(A)tau f(B)right),end{align*}for the matrix means $sigma,tauin{nabla_{alpha},!_{alpha}}$ and $alphain[0,1]$. Further, if $sigma=sharp_{alpha}$, thenbegin{align*} lambda^downarrow left(fleft(e^{Anabla_{alpha}B}right)right)prec_wlambda^downarrow left(f(e^A)tau f(e^B))right).end{align*}Similar inequalities will be presented for two-variable functions too.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43631402","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}
F. S. Dahlgren, Zachary Gershkoff, L. Hogben, S. Motlaghian, Derek Young
A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.
{"title":"Inverse eigenvalue and related problems for hollow matrices described by graphs","authors":"F. S. Dahlgren, Zachary Gershkoff, L. Hogben, S. Motlaghian, Derek Young","doi":"10.13001/ela.2022.6941","DOIUrl":"https://doi.org/10.13001/ela.2022.6941","url":null,"abstract":"A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46380290","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 Laplacian-energy-like invariant and the Kirchhoff index of an $n$-vertex simple connected graph $G$ are, respectively, defined to be $LEL(G)=sum_{i=1}^{n-1}sqrt{mu_i}$ and $Kf(G)=nsum_{i=1}^{n-1}frac{1}{mu_i}$, where $mu_1,mu_2,ldots,mu_{n-1},mu_n=0$ are the Laplacian eigenvalues of $G$. In this paper, some results in the paper [Comparison between the Laplacian-energy-like invariant and the Kirchhoff index. Electron. J. Linear Algebra 31:27-41, 2016] are corrected and improved.
{"title":"Remarks on \"Comparison between the Laplacian energy-like invariant and the Kirchhoff index''","authors":"Xiaodan Chen, Guoliang Hao","doi":"10.13001/ela.2022.6383","DOIUrl":"https://doi.org/10.13001/ela.2022.6383","url":null,"abstract":"The Laplacian-energy-like invariant and the Kirchhoff index of an $n$-vertex simple connected graph $G$ are, respectively, defined to be $LEL(G)=sum_{i=1}^{n-1}sqrt{mu_i}$ and $Kf(G)=nsum_{i=1}^{n-1}frac{1}{mu_i}$, where $mu_1,mu_2,ldots,mu_{n-1},mu_n=0$ are the Laplacian eigenvalues of $G$. In this paper, some results in the paper [Comparison between the Laplacian-energy-like invariant and the Kirchhoff index. Electron. J. Linear Algebra 31:27-41, 2016] are corrected and improved.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42690587","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 algebraic connectivity of a connected graph $G$ is the second smallest eigenvalue of the Laplacian matrix of $G$. In this paper, some new upper bounds on algebraic connectivity are obtained by applying generalized interlacing to an appropriate quotient matrix.
{"title":"Upper bounds on the algebraic connectivity of graphs","authors":"Zhen Lin, L. Miao","doi":"10.13001/ela.2022.5133","DOIUrl":"https://doi.org/10.13001/ela.2022.5133","url":null,"abstract":"The algebraic connectivity of a connected graph $G$ is the second smallest eigenvalue of the Laplacian matrix of $G$. In this paper, some new upper bounds on algebraic connectivity are obtained by applying generalized interlacing to an appropriate quotient matrix.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46460047","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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