Sign-restricted matrices (SRMs) are $(0, pm 1)$-matrices where, ignoring 0's, the signs in each column alternate beginning with a $+1$ and all partial row sums are nonnegative. The most investigated of these matrices are the alternating sign matrices (ASMs), where the rows also have the alternating sign property, and all row and column sums equal 1. We introduce monotone triangles to represent SRMs and investigate some of their properties and connections to certain polytopes. We also investigate two partial orders for ASMs related to their patterns alternating cycles and show a number of combinatorial properties of these orders.
{"title":"Alternating sign and sign-restricted matrices: representations and partial orders","authors":"R. Brualdi, G. Dahl","doi":"10.13001/ela.2021.6513","DOIUrl":"https://doi.org/10.13001/ela.2021.6513","url":null,"abstract":"Sign-restricted matrices (SRMs) are $(0, pm 1)$-matrices where, ignoring 0's, the signs in each column alternate beginning with a $+1$ and all partial row sums are nonnegative. The most investigated of these matrices are the alternating sign matrices (ASMs), where the rows also have the alternating sign property, and all row and column sums equal 1. We introduce monotone triangles to represent SRMs and investigate some of their properties and connections to certain polytopes. We also investigate two partial orders for ASMs related to their patterns alternating cycles and show a number of combinatorial properties of these orders.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46211668","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 data science and machine learning, the method of nonnegative matrix factorization (NMF) is a powerful tool that enjoys great popularity. Depending on the concrete application, there exist several subclasses each of which performs a NMF under certain constraints. Consider a given square matrix $A$. The symmetric NMF aims for a nonnegative low-rank approximation $Aapprox XX^T$ to $A$, where $X$ is entrywise nonnegative and of given order. Considering a rectangular input matrix $A$, the general NMF again aims for a nonnegative low-rank approximation to $A$ which is now of the type $Aapprox XY$ for entrywise nonnegative matrices $X,Y$ of given order. In this paper, we introduce a new heuristic method to tackle the exact nonnegative matrix factorization problem (of type $A=XY$), based on projection approaches to solve a certain feasibility problem.
在数据科学和机器学习中，非负矩阵分解(NMF)方法是一个非常受欢迎的强大工具。根据具体的应用，存在几个子类，每个子类在特定的约束下执行NMF。考虑一个给定的方阵a。对称NMF的目标是一个非负的低秩近似$ a 约XX^T$到$ a $，其中$X$是非负的并且是给定顺序的。考虑一个矩形输入矩阵$ a $，一般的NMF再次以$ a $的非负低秩近似为目标，对于给定顺序的入口非负矩阵$X,Y$，它现在的类型为$ a 约XY$。本文引入了一种新的启发式方法来解决(类型为$ a =XY$)的精确非负矩阵分解问题，该方法基于求解某可行性问题的投影方法。
{"title":"A projective approach to nonnegative matrix factorization","authors":"Patrick Groetzner","doi":"10.13001/ela.2021.5067","DOIUrl":"https://doi.org/10.13001/ela.2021.5067","url":null,"abstract":"In data science and machine learning, the method of nonnegative matrix factorization (NMF) is a powerful tool that enjoys great popularity. Depending on the concrete application, there exist several subclasses each of which performs a NMF under certain constraints. Consider a given square matrix $A$. The symmetric NMF aims for a nonnegative low-rank approximation $Aapprox XX^T$ to $A$, where $X$ is entrywise nonnegative and of given order. Considering a rectangular input matrix $A$, the general NMF again aims for a nonnegative low-rank approximation to $A$ which is now of the type $Aapprox XY$ for entrywise nonnegative matrices $X,Y$ of given order. In this paper, we introduce a new heuristic method to tackle the exact nonnegative matrix factorization problem (of type $A=XY$), based on projection approaches to solve a certain feasibility problem.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46321596","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}
N. Johnston, Shirin Moein, Rajesh Pereira, S. Plosker
Numerous results are presented that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices is developed. Applications in the theory of quantum resources are explored. For example, the quantum states that have modified trace distance of coherence equal to $1$ (the maximal possible value) are characterized, and a connection between the modified trace distance of $2$-entanglement and the NPPT bound entanglement problem is established.
{"title":"Birkhoff-James orthogonality in the trace norm, with applications to quantum resource theories","authors":"N. Johnston, Shirin Moein, Rajesh Pereira, S. Plosker","doi":"10.13001/ela.2022.7149","DOIUrl":"https://doi.org/10.13001/ela.2022.7149","url":null,"abstract":"Numerous results are presented that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices is developed. Applications in the theory of quantum resources are explored. For example, the quantum states that have modified trace distance of coherence equal to $1$ (the maximal possible value) are characterized, and a connection between the modified trace distance of $2$-entanglement and the NPPT bound entanglement problem is established.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42387240","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}
Gau, Wang and Wu in their LAMA'2016 paper conjectured (and proved for $nleq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds for $n=5$.
{"title":"The Gau-Wang-Wu conjecture on partial isometries holds in the 5-by-5 case","authors":"I. Spitkovsky, I. Suleiman, E. Wegert","doi":"10.13001/ela.2022.6541","DOIUrl":"https://doi.org/10.13001/ela.2022.6541","url":null,"abstract":"Gau, Wang and Wu in their LAMA'2016 paper conjectured (and proved for $nleq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds for $n=5$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47457292","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 central mathematical problem studied in this work is the estimation of the quadratic form $x^TA^{-1}x$ for a given symmetric positive definite matrix $A in mathbb{R}^{n times n}$ and vector $x in mathbb{R}^n$. Several methods to estimate $x^TA^{-1}x$ without computing the matrix inverse are proposed. The precision of the estimates is analyzed both analytically and numerically.
{"title":"On the estimation of ${x}^TA^{-1}{x}$ for symmetric matrices","authors":"Paraskevi Fika, M. Mitrouli, Ondrej Turec","doi":"10.13001/ela.2021.5611","DOIUrl":"https://doi.org/10.13001/ela.2021.5611","url":null,"abstract":"The central mathematical problem studied in this work is the estimation of the quadratic form $x^TA^{-1}x$ for a given symmetric positive definite matrix $A in mathbb{R}^{n times n}$ and vector $x in mathbb{R}^n$. Several methods to estimate $x^TA^{-1}x$ without computing the matrix inverse are proposed. The precision of the estimates is analyzed both analytically and numerically. \u0000 ","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49543038","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 present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear tensor equations. We use the T-product for two tensors to define tensor tubal global Arnoldi and tensor tubal global Golub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based global approaches can be exploited to solve ill-posed problems arising from recovering blurry multichannel (color) images and videos, using the so-called Tikhonov regularization technique, to provide computable approximate regularized solutions. We also review a generalized cross-validation and discrepancy principle type of criterion for the selection of the regularization parameter in the Tikhonov regularization. Applications to image sequence processing are given to demonstrate the efficiency of the algorithms.
{"title":"On tensor GMRES and Golub-Kahan methods via the T-product for color image processing","authors":"M. E. Guide, Alaa El Ichi, K. Jbilou, R. Sadaka","doi":"10.13001/ELA.2021.5471","DOIUrl":"https://doi.org/10.13001/ELA.2021.5471","url":null,"abstract":"The present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear tensor equations. We use the T-product for two tensors to define tensor tubal global Arnoldi and tensor tubal global Golub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based global approaches can be exploited to solve ill-posed problems arising from recovering blurry multichannel (color) images and videos, using the so-called Tikhonov regularization technique, to provide computable approximate regularized solutions. We also review a generalized cross-validation and discrepancy principle type of criterion for the selection of the regularization parameter in the Tikhonov regularization. Applications to image sequence processing are given to demonstrate the efficiency of the algorithms.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41383661","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 propose an error analysis of the generalized low-rank approximation, which is a generalization of the classical approximation of a matrix $Ainmathbb{R}^{mtimes n}$ by a matrix of a rank at most $r$, where $rleqmin{m,n}$.
{"title":"Error analysis of the generalized low-rank matrix approximation","authors":"Pablo Soto-Quiros","doi":"10.13001/ELA.2021.5961","DOIUrl":"https://doi.org/10.13001/ELA.2021.5961","url":null,"abstract":"In this paper, we propose an error analysis of the generalized low-rank approximation, which is a generalization of the classical approximation of a matrix $Ainmathbb{R}^{mtimes n}$ by a matrix of a rank at most $r$, where $rleqmin{m,n}$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41572184","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}
Given free modules $Msubseteq L$ of finite rank $fgeq 1$ over a principal ideal domain $R$, we give a procedure to construct a basis of $L$ from a basis of $M$ assuming the invariant factors or elementary divisors of $L/M$ are known. Given a matrix $Ain M_{m,n}(R)$ of rank $r$, its nullspace $L$ in $R^n$ is a free $R$-module of rank $f=n-r$. We construct a free submodule $M$ of $L$ of rank $f$ naturally associated with $A$ and whose basis is easily computable, we determine the invariant factors of the quotient module $L/M$ and then indicate how to apply the previous procedure to build a basis of $L$ from one of $M$.
{"title":"Linear systems of Diophantine equations","authors":"F. Szechtman","doi":"10.13001/ela.2022.6695","DOIUrl":"https://doi.org/10.13001/ela.2022.6695","url":null,"abstract":"Given free modules $Msubseteq L$ of finite rank $fgeq 1$ over a principal ideal domain $R$, we give a procedure to construct a basis of $L$ from a basis of $M$ assuming the invariant factors or elementary divisors of $L/M$ are known. Given a matrix $Ain M_{m,n}(R)$ of rank $r$, its nullspace $L$ in $R^n$ is a free $R$-module of rank $f=n-r$. We construct a free submodule $M$ of $L$ of rank $f$ naturally associated with $A$ and whose basis is easily computable, we determine the invariant factors of the quotient module $L/M$ and then indicate how to apply the previous procedure to build a basis of $L$ from one of $M$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43873466","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}
G. Groenewald, D.B. Janse van Rensburg, A. Ran, F. Theron, M. Van Straaten
Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an $H$-polar decomposition are found. In the process, an equivalent to Witt's theorem on extending $H$-isometries to $H$-unitary matrices is given for quaternion matrices.
{"title":"Polar decompositions of quaternion matrices in indefinite inner product spaces","authors":"G. Groenewald, D.B. Janse van Rensburg, A. Ran, F. Theron, M. Van Straaten","doi":"10.13001/ela.2021.6411","DOIUrl":"https://doi.org/10.13001/ela.2021.6411","url":null,"abstract":"Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an $H$-polar decomposition are found. In the process, an equivalent to Witt's theorem on extending $H$-isometries to $H$-unitary matrices is given for quaternion matrices.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42025306","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 tensor rank and border rank of the $3 times 3$ determinant tensor are known to be $5$ if the characteristic is not two. In characteristic two, the existing proofs of both the upper and lower bounds fail. In this paper, we show that the tensor rank remains $5$ for fields of characteristic two as well.
{"title":"On the tensor rank of the 3 x 3 permanent and determinant","authors":"Siddharth Krishna, V. Makam","doi":"10.13001/ELA.2021.5107","DOIUrl":"https://doi.org/10.13001/ELA.2021.5107","url":null,"abstract":"The tensor rank and border rank of the $3 times 3$ determinant tensor are known to be $5$ if the characteristic is not two. In characteristic two, the existing proofs of both the upper and lower bounds fail. In this paper, we show that the tensor rank remains $5$ for fields of characteristic two as well.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88182354","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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