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 Majorization inequalities via convex functions <p>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. <br>For example, it is shown that if $A,B$ are positive definite matrices and $f$ is a continuous $\sigma\tau$-convex function on an interval containing the spectra of $A,B$, then<br>\begin{align*}<br>\lambda^\downarrow (f(A\sigma B))\prec_w\lambda^\downarrow \left(f(A)\tau f(B)\right),<br>\end{align*}<br>for the matrix means $\sigma,\tau\in\{\nabla_{\alpha},!_{\alpha}\}$ and $\alpha\in[0,1]$. Further, if $\sigma=\sharp_{\alpha}$, then<br>\begin{align*} \lambda^\downarrow \left(f\left(e^{A\nabla_{\alpha}B}\right)\right)\prec_w\lambda^\downarrow \left(f(e^A)\tau f(e^B))\right).<br>\end{align*}<br>Similar inequalities will be presented for two-variable functions too.</p> Mohsen Kian Mohammad Sababheh Copyright (c) 2022 Mohsen Kian, Mohammad Sababheh 2022-02-22 2022-02-22 179 194 10.13001/ela.2022.6901 On the structure of isometrically embeddable metric spaces <p>Since its popularization in the 1970s, the Fiedler vector of a graph has become a standard tool for clustering of the vertices of the graph. Recently, Mendel and Noar, Dumitriu and Radcliffe, and Radcliffe and Williamson have introduced geometric generalizations of the Fiedler vector. Motivated by questions stemming from their work, we provide structural characterizations for when a finite metric space can be isometrically embedded in a Hilbert space.</p> Kathleen Nowak Carlos Ortiz Marrero Stephen Young Copyright (c) 2022-02-17 2022-02-17 91 106 10.13001/ela.2022.6891 Sign patterns of rational matrices with large rank II <p>It is known that, for any real m-by-n matrix A of rank n-2, there is a rational m-by-n matrix which has rank n-2 and sign pattern equal to that of&nbsp; A. We prove a more general result conjectured in the recent literature.</p> <p>&nbsp;</p> Yaroslav Shitov Copyright (c) 2022-01-12 2022-01-12 63 64 10.13001/ela.2022.6807 On decompositions of matrices into products of commutators of involutions <p>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.</p> Tran Nam Son Truong Huu Dung Nguyen Thi Thai Ha Mai Hoang Bien Copyright (c) 2022-02-23 2022-02-23 123 130 10.13001/ela.2022.6797 Constructions of cospectral graphs with different zero forcing numbers <p>Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper, we show that several NP-hard zero forcing numbers are not characterized by the spectra of several types of associated matrices with a graph. In particular, we consider standard zero forcing, positive semidefinite zero forcing, and skew zero forcing and provide constructions of infinite families of pairs of cospectral graphs, which have different values for these numbers. We explore several methods for obtaining these cospectral graphs including using graph products, graph joins, and graph switching. Among these, we provide a construction involving regular adjacency cospectral graphs; the regularity of this construction also implies cospectrality with respect to several other matrices including the Laplacian, signless Laplacian, and normalized Laplacian. We also provide a construction where pairs of cospectral graphs can have an arbitrarily large difference between their zero forcing numbers.</p> Aida Abiad Boris Brimkov Jane Breen Thomas R. Cameron Himanshu Gupta Ralihe R. Villagran Copyright (c) 2022 Aida Abiad, Boris Brimkov, Jane Breen, Thomas R. Cameron, Himanshu Gupta, Ralihe R. Villagran 2022-05-05 2022-05-05 280 294 10.13001/ela.2022.6737