The Electronic Journal of Linear Algebra <p>The Electronic Journal of Linear Algebra (ELA), a publication of the <a href="">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.&nbsp;ELA is a JCR ranked journal, and indexed by MathSciNet, ZentralBlatt, and Scopus. &nbsp;ELA is completely free for authors/readers; and &nbsp;ELA is built on the selfless contributions of its authors, referees and editors.</p> <p><a href="" target="_blank" rel="noopener"><img src="" alt="" width="62" height="62"></a></p> International Linear Algebra Society en-US The Electronic Journal of Linear Algebra 1081-3810 A projective approach to nonnegative matrix factorization <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 2021-09-13 2021-09-13 37 583 597 10.13001/ela.2021.5067 Nonparallel flat portions on the boundaries of numerical ranges of 4-by-4 nilpotent matrices <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 Cox Weston Grewe Grace Hochrein Linda Patton Ilya Spitkovsky Copyright (c) 2021 Mackenzie Cox, Weston Grewe, Grace Hochrein, Linda Patton, Ilya Spitkovsky 2021-08-03 2021-08-03 37 504 523 10.13001/ela.2021.5785 Potentially stable and 5-by-5 spectrally arbitrary tree sign pattern matrices with all edges negative <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 2021-07-29 2021-07-29 37 562 582 10.13001/ela.2021.5969 On the estimation of ${x}^TA^{-1}{x}$ for symmetric matrices <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>&nbsp;</p> Paraskevi Fika Marilena Mitrouli Ondrej Turek Copyright (c) 2021 Paraskevi Fika, Marilena Mitrouli, Ondrej Turec 2021-07-28 2021-07-28 37 549 561 10.13001/ela.2021.5611 Error analysis of the generalized low-rank matrix approximation <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 2021-07-23 2021-07-23 37 544 548 10.13001/ela.2021.5961