An $n$-by-$n$ matrix $A$ is called symmetric, skew-symmetric, and orthogonal if $A^T=A$, $A^T=-A$, and $A^T=A^{-1}$, respectively. We give necessary and sufficient conditions on a complex matrix $A$ so that it is a sum of type ``"orthogonal $+$ symmetric" in terms of the Jordan form of $A-A^T$. We also give necessary and sufficient conditions on a complex matrix $A$ so that it is a sum of type "orthogonal $+$ skew-symmetric" in terms of the Jordan form of $A+A^T$.
如果$A^T=A$、$A^T=-A$和$A^T=A^{-1}$，则一个$n$ × $n$矩阵$A$分别称为对称、偏对称和正交矩阵$A$。给出了复矩阵$ a $在$ a - a ^T$的约当形式下是“正交$+对称$”型和的充要条件。我们还给出了复矩阵$ a $在$ a + a ^T$的约当形式下是“正交$+$偏对称”型和的充要条件。
{"title":"Sums of orthogonal, symmetric, and skew-symmetric matrices","authors":"Ralph John de la Cruz, Agnes T. Paras","doi":"10.13001/ela.2022.7129","DOIUrl":"https://doi.org/10.13001/ela.2022.7129","url":null,"abstract":"An $n$-by-$n$ matrix $A$ is called symmetric, skew-symmetric, and orthogonal if $A^T=A$, $A^T=-A$, and $A^T=A^{-1}$, respectively. We give necessary and sufficient conditions on a complex matrix $A$ so that it is a sum of type ``\"orthogonal $+$ symmetric\" in terms of the Jordan form of $A-A^T$. We also give necessary and sufficient conditions on a complex matrix $A$ so that it is a sum of type \"orthogonal $+$ skew-symmetric\" in terms of the Jordan form of $A+A^T$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45055727","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 new class of directed trees is introduced. A formula for the group inverse of the matrices associated with any tree belonging to this class is obtained. This answers affirmatively, a conjecture of Catral et al., for this new class.
{"title":"Group inverses of matrices of directed trees","authors":"R. Nandi, K. Sivakumar","doi":"10.13001/ela.2022.7093","DOIUrl":"https://doi.org/10.13001/ela.2022.7093","url":null,"abstract":"A new class of directed trees is introduced. A formula for the group inverse of the matrices associated with any tree belonging to this class is obtained. This answers affirmatively, a conjecture of Catral et al., for this new class.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48205040","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 article, we introduce two new generalized inverses for rectangular matrices called $W$-weighted generalized-Drazin--Moore--Penrose (GDMP) and $W$-weighted generalized-Drazin-reflexive (GDR) inverses. The first generalized inverse can be seen as a generalization of the recently introduced GDMP inverse for a square matrix to a rectangular matrix. The second class of generalized inverse contains the class of the first generalized inverse. We then exploit their various properties and establish that the proposed generalized inverses coincide with different well-known generalized inverses under certain assumptions. We also obtain a representation of $W$-weighted GDMP inverse employing EP-core nilpotent decomposition. We define the dual of $W$-weighted GDMP inverse and obtain analogue results. Further, we discuss additive properties, reverse- and forward-order laws for GD, $W$-weighted GD, GDMP, and $W$-weighted GDMP generalized inverses.
{"title":"W-weighted GDMP inverse for rectangular matrices","authors":"Amit Kumar, Vaibhav Shekhar, Debasisha Mishra","doi":"10.13001/ela.2022.7015","DOIUrl":"https://doi.org/10.13001/ela.2022.7015","url":null,"abstract":"In this article, we introduce two new generalized inverses for rectangular matrices called $W$-weighted generalized-Drazin--Moore--Penrose (GDMP) and $W$-weighted generalized-Drazin-reflexive (GDR) inverses. The first generalized inverse can be seen as a generalization of the recently introduced GDMP inverse for a square matrix to a rectangular matrix. The second class of generalized inverse contains the class of the first generalized inverse. We then exploit their various properties and establish that the proposed generalized inverses coincide with different well-known generalized inverses under certain assumptions. We also obtain a representation of $W$-weighted GDMP inverse employing EP-core nilpotent decomposition. We define the dual of $W$-weighted GDMP inverse and obtain analogue results. Further, we discuss additive properties, reverse- and forward-order laws for GD, $W$-weighted GD, GDMP, and $W$-weighted GDMP generalized inverses.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46166370","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}
Milica Andelic, Carlos M. da Fonseca, E. Kılıç, Z. Stanić
In this paper, we provide a new family of tridiagonal matrices whose eigenvalues are perfect squares. This result motivates the computation of the spectrum of a particular antibidiagonal matrix. As an application, we consider the Laplacian controllability of a particular subclass of chain graphs known as half graphs.
{"title":"A Sylvester-Kac matrix type and the Laplacian controllability of half graphs","authors":"Milica Andelic, Carlos M. da Fonseca, E. Kılıç, Z. Stanić","doi":"10.13001/ela.2022.6947","DOIUrl":"https://doi.org/10.13001/ela.2022.6947","url":null,"abstract":"In this paper, we provide a new family of tridiagonal matrices whose eigenvalues are perfect squares. This result motivates the computation of the spectrum of a particular antibidiagonal matrix. As an application, we consider the Laplacian controllability of a particular subclass of chain graphs known as half graphs.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46298065","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 class of unicyclic $3$-colored digraphs with the cycle weight $pmmathrm{i}$ and with a unique perfect matching, denoted by $mathcal{U}_g$, is considered in this article. Kalita & Sarma [On the inverse of unicyclic 3-coloured digraphs, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2021.1948956] introduced the notion of inverse of $3$-colored digraphs. They characterized the unicyclic $3$-colored digraphs in $mathcal{U}_g$ possessing unicyclic inverses. This article provides a complete characterization of the unicyclic $3$-colored digraphs in $mathcal{U}_g$ possessing bicyclic inverses.
{"title":"Unicyclic 3-colored digraphs with bicyclic inverses","authors":"D. Kalita, K. Sarma","doi":"10.13001/ela.2022.7037","DOIUrl":"https://doi.org/10.13001/ela.2022.7037","url":null,"abstract":"The class of unicyclic $3$-colored digraphs with the cycle weight $pmmathrm{i}$ and with a unique perfect matching, denoted by $mathcal{U}_g$, is considered in this article. Kalita & Sarma [On the inverse of unicyclic 3-coloured digraphs, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2021.1948956] introduced the notion of inverse of $3$-colored digraphs. They characterized the unicyclic $3$-colored digraphs in $mathcal{U}_g$ possessing unicyclic inverses. This article provides a complete characterization of the unicyclic $3$-colored digraphs in $mathcal{U}_g$ possessing bicyclic inverses.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43994716","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 aim of this paper is to analyze the asymptotic behavior of the eigenvalues and eigenvectors of particular sequences of products involving two square real matrices $A$ and $B$, namely of the form $B^kA$, as $krightarrow infty$. This analysis represents a detailed deepening of a particular case within a general theory on finite families $mathcal{F} = { A_1, ldots, A_m }$ of real square matrices already available in the literature. The Bachmann-Landau symbols and related results are largely used and are presented in a systematic way in the final Appendix.
{"title":"Spectral properties of certain sequences of products of two real matrices","authors":"M. Brundu, M. Zennaro","doi":"10.13001/ela.2022.6651","DOIUrl":"https://doi.org/10.13001/ela.2022.6651","url":null,"abstract":"\u0000\u0000\u0000The aim of this paper is to analyze the asymptotic behavior of the eigenvalues and eigenvectors of particular sequences of products involving two square real matrices $A$ and $B$, namely of the form $B^kA$, as $krightarrow infty$. This analysis represents a detailed deepening of a particular case within a general theory on finite families $mathcal{F} = { A_1, ldots, A_m }$ of real square matrices already available in the literature. The Bachmann-Landau symbols and related results are largely used and are presented in a systematic way in the final Appendix.\u0000\u0000\u0000","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":"70 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41289841","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 give necessary and sufficient conditions for a real symmetric matrix and, in particular, for the distance matrix $D(H_n)$ of a helm graph $H_n$ to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank-one matrix. As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86-104, 2021), for $D(H_n)$ when $n$ is even. Further, we derive a formula for the Moore-Penrose inverse of singular $D(H_n)$ that is analogous to the formula for $D(H_n)^{-1}$. Precisely, if $n$ is odd, we find a symmetric positive semi-definite Laplacian-like matrix $L$ of order $2n-1$ and a vector $mathbf{w}in mathbb{R}^{2n-1}$ such thatbegin{eqnarray*}D(H_n)^{dagger} = -frac{1}{2}L +frac{4}{3(n-1)}mathbf{w}mathbf{w^{prime}},end{eqnarray*}where the rank of $L$ is $2n-3$. We also investigate the inertia of $D(H_n)$.
{"title":"The Moore-Penrose inverse of the distance matrix of a helm graph","authors":"I. Jeyaraman, T. Divyadevi, R. Azhagendran","doi":"10.13001/ela.2023.7465","DOIUrl":"https://doi.org/10.13001/ela.2023.7465","url":null,"abstract":"In this paper, we give necessary and sufficient conditions for a real symmetric matrix and, in particular, for the distance matrix $D(H_n)$ of a helm graph $H_n$ to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank-one matrix. As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86-104, 2021), for $D(H_n)$ when $n$ is even. Further, we derive a formula for the Moore-Penrose inverse of singular $D(H_n)$ that is analogous to the formula for $D(H_n)^{-1}$. Precisely, if $n$ is odd, we find a symmetric positive semi-definite Laplacian-like matrix $L$ of order $2n-1$ and a vector $mathbf{w}in mathbb{R}^{2n-1}$ such thatbegin{eqnarray*}D(H_n)^{dagger} = -frac{1}{2}L +frac{4}{3(n-1)}mathbf{w}mathbf{w^{prime}},end{eqnarray*}where the rank of $L$ is $2n-3$. We also investigate the inertia of $D(H_n)$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44270804","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 square matrix $A$ is an involution if $A^{2} = I$. The centralizer of a square matrix $S$ denoted by $mathscr{C}(S)$ is the set of all $A$ such that $AS = SA$ over an algebraically closed field of characteristic not equal to 2. We determine necessary and sufficient conditions for $A in mathscr{C}(S)$ to be a product of involutions in $mathscr{C}(S)$ where $S$ is a basic Weyr matrix with homogeneous Weyr structure of length 3. Finally, we will show some results for the case when the length of the Weyr structure is greater than 3.
一个方阵$A$是一个对合矩阵，如果$A^{2} = I$。用$mathscr{C}(S)$表示的方阵$S$的中心化器是在特征不等于2的代数闭域上满足$AS = SA$的所有$ a $的集合。我们确定了$A in mathscr{C}(S)$是$mathscr{C}(S)$的对合积的充要条件，其中$S$是一个长度为3的齐次Weyr结构的基本Weyr矩阵。最后，我们将给出Weyr结构长度大于3时的一些结果。
{"title":"The products of involutions in a matrix centralizer","authors":"Ralph John de la Cruz, Raymond Louis Tañedo","doi":"10.13001/ela.2022.7091","DOIUrl":"https://doi.org/10.13001/ela.2022.7091","url":null,"abstract":"A square matrix $A$ is an involution if $A^{2} = I$. The centralizer of a square matrix $S$ denoted by $mathscr{C}(S)$ is the set of all $A$ such that $AS = SA$ over an algebraically closed field of characteristic not equal to 2. We determine necessary and sufficient conditions for $A in mathscr{C}(S)$ to be a product of involutions in $mathscr{C}(S)$ where $S$ is a basic Weyr matrix with homogeneous Weyr structure of length 3. Finally, we will show some results for the case when the length of the Weyr structure is greater than 3.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41919464","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}
For an $ntimes n$ matrix $M$, $sigma(M)$ denotes the set of all different eigenvalues of $M$. In this paper, we will prove two results on the $m$-th $(mgeq2)$ roots of a matrix $A$. Firstly, let $X$ be an $m$-th root of $A$. Then $X$ can be expressed as a polynomial in $A$ if and only if rank $X^2$= rank $X$ and $|sigma(X)|=|sigma(A)|$. Secondly, let $X$ and $Y$ be two $m$-th roots of $A$. If both $X$ and $Y$ can be expressed as polynomials in $A$, then $X=Y$ if and only if $sigma(X)=sigma(Y)$.
{"title":"On m-th roots of complex matrices","authors":"H. Liu, Jing Zhao","doi":"10.13001/ela.2022.7047","DOIUrl":"https://doi.org/10.13001/ela.2022.7047","url":null,"abstract":"For an $ntimes n$ matrix $M$, $sigma(M)$ denotes the set of all different eigenvalues of $M$. In this paper, we will prove two results on the $m$-th $(mgeq2)$ roots of a matrix $A$. Firstly, let $X$ be an $m$-th root of $A$. Then $X$ can be expressed as a polynomial in $A$ if and only if rank $X^2$= rank $X$ and $|sigma(X)|=|sigma(A)|$. Secondly, let $X$ and $Y$ be two $m$-th roots of $A$. If both $X$ and $Y$ can be expressed as polynomials in $A$, then $X=Y$ if and only if $sigma(X)=sigma(Y)$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44485991","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 counterexample to a theorem in the paper ELA 29:3-16, (2015) is provided, and an upper bound on the H-spectral radius of H-tensors is given.
本文给出了ELA 29:3-16，(2015)中一个定理的反例，并给出了h张量的h谱半径的上界。
{"title":"A note on bounds for eigenvalues of nonsingular H-tensors","authors":"Jun He, Guanjun Xu","doi":"10.13001/ela.2022.7097","DOIUrl":"https://doi.org/10.13001/ela.2022.7097","url":null,"abstract":"A counterexample to a theorem in the paper ELA 29:3-16, (2015) is provided, and an upper bound on the H-spectral radius of H-tensors is given.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44861301","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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