The Electronic Journal of Linear Algebra
https://journals.uwyo.edu/index.php/ela
<p>The Electronic Journal of Linear Algebra (ELA), a publication of the <a href="https://www.ilasic.org/">International Linear Algebra Society (ILAS)</a>, is a refereed all-electronic journal that welcomes mathematical articles of high standards that contribute new information and new insights to matrix analysis and the various aspects of linear algebra and its applications. ELA is a JCR ranked journal, and indexed by MathSciNet, ZentralBlatt, and Scopus. ELA is completely free for authors/readers; and ELA is built on the selfless contributions of its authors, referees and editors.</p> <p><a href="https://en.wikipedia.org/wiki/Electronic_Journal_of_Linear_Algebra" target="_blank" rel="noopener"><img src="https://journals.uwyo.edu/public/site/images/dopico/wikipedia.png" alt="" width="62" height="62"></a></p>International Linear Algebra Societyen-USThe Electronic Journal of Linear Algebra1081-3810A projective approach to nonnegative matrix factorization
https://journals.uwyo.edu/index.php/ela/article/view/5067
<p>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 $A\approx XX^T$ 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 $A\approx XY$ 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.</p>Patrick Groetzner
Copyright (c) 2021 Patrick Groetzner
2021-09-132021-09-133758359710.13001/ela.2021.5067Nonparallel flat portions on the boundaries of numerical ranges of 4-by-4 nilpotent matrices
https://journals.uwyo.edu/index.php/ela/article/view/5785
<p>The 4-by-4 nilpotent matrices whose numerical ranges have nonparallel flat portions on their boundary that are on lines equidistant from the origin are characterized. Their numerical ranges are always symmetric about a line through the origin and all possible angles between the lines containing the flat portions are attained.</p>Mackenzie CoxWeston GreweGrace HochreinLinda PattonIlya Spitkovsky
Copyright (c) 2021 Mackenzie Cox, Weston Grewe, Grace Hochrein, Linda Patton, Ilya Spitkovsky
2021-08-032021-08-033750452310.13001/ela.2021.5785Potentially stable and 5-by-5 spectrally arbitrary tree sign pattern matrices with all edges negative
https://journals.uwyo.edu/index.php/ela/article/view/5969
<p>Characterization of potentially stable sign pattern matrices has been a long-standing open problem. In this paper, we give some sufficient conditions for tree sign pattern matrices with all edges negative to allow a properly signed nest. We also characterize potentially stable star and path sign pattern matrices with all edges negative. We give a conjecture on characterizing potentially stable tree sign pattern matrices with all edges negative in terms of allowing a properly signed nest which is verified to be true for sign pattern matrices up to order 6. Finally, we characterize all 5-by-5 spectrally arbitrary tree sign pattern matrices with all edges negative.</p>Sunil Das
Copyright (c) 2021 Sunil Das
2021-07-292021-07-293756258210.13001/ela.2021.5969On the estimation of ${x}^TA^{-1}{x}$ for symmetric matrices
https://journals.uwyo.edu/index.php/ela/article/view/5611
<p>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.</p> <p> </p>Paraskevi FikaMarilena MitrouliOndrej Turek
Copyright (c) 2021 Paraskevi Fika, Marilena Mitrouli, Ondrej Turec
2021-07-282021-07-283754956110.13001/ela.2021.5611Error analysis of the generalized low-rank matrix approximation
https://journals.uwyo.edu/index.php/ela/article/view/5961
<p>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 $A\in\mathbb{R}^{m\times n}$ by a matrix of a rank at most $r$, where $r\leq\min\{m,n\}$.</p>Pablo Soto-Quiros
Copyright (c) 2021 Pablo Soto-Quiros
2021-07-232021-07-233754454810.13001/ela.2021.5961