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.&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> International Linear Algebra Society en-US The Electronic Journal of Linear Algebra 1081-3810 Extreme Points of Certain Transportation Polytopes with Fixed Total Sums https://journals.uwyo.edu/index.php/ela/article/view/5141 <p>Transportation matrices are $m\times n$ nonnegative matrices with given row sum vector $R$ and column sum vector $S$. All such matrices form the convex polytope $\mathcal{U}(R,S)$ which is called a transportation polytope and its extreme points have been classified. In this article, we consider a new class of convex polytopes $\Delta(\bar{R},\bar{S},\sigma)$ consisting of certain transportation polytopes satisfying that the sum of all elements is $\sigma$, and the row and column sum vectors are dominated componentwise by the given positive vectors $\bar{R}$ and $\bar{S}$, respectively. We characterize the extreme points of $\Delta(\bar{R},\bar{S},\sigma)$. Moreover, we give the minimal term rank and maximal permanent of $\Delta(\bar{R},\bar{S},\sigma)$.</p> Zhi Chen Zelin Zhu Jiawei Li Lizhen Yang Lei Cao Copyright (c) 2021-04-05 2021-04-05 37 256 271 10.13001/ela.2021.5141 m-commuting maps on triangular and strictly triangular infinite matrices https://journals.uwyo.edu/index.php/ela/article/view/5083 <p>Let $N_\infty(F)$ be the ring of infinite strictly upper triangular matrices with entries in an infinite field. The description of the commuting maps defined on $N_\infty(F)$, i.e. the maps $f\colon N_\infty(F)\rightarrow N_\infty(F)$ such that $[f(X),X]=0$ for every $X\in N_\infty(F)$, is presented. With the use of this result, the form of $m$-commuting maps defined on $T_\infty(F)$ -- the ring of infinite upper triangular matrices, i.e. the maps $f\colon T_\infty(F)\rightarrow T_\infty(F)$ such that $[f(X),X^m]=0$ for every $X\in T_\infty(F)$, is found.</p> Roksana SÅ‚owik Driss Aiat Hadj Ahmed Copyright (c) 2021-03-24 2021-03-24 37 247 255 10.13001/ela.2021.5083 Spectral theory for self-adjoint quadratic eigenvalue problems - a review https://journals.uwyo.edu/index.php/ela/article/view/5361 <p>Many physical problems require the spectral analysis of quadratic matrix polynomials $M\lambda^2+D\lambda +K$, $\lambda \in \mathbb{C}$, with $n \times n$ Hermitian matrix coefficients, $M,\;D,\;K$. In this largely expository paper, we present and discuss canonical forms for these polynomials under the action of both congruence and similarity transformations of a linearization and also $\lambda$-dependent unitary similarity transformations of the polynomial itself. Canonical structures for these processes are clarified, with no restrictions on eigenvalue multiplicities. Thus, we bring together two lines of attack: (a) <strong>analytic</strong> via direct reduction of the $n \times n$ system itself by $\lambda$-dependent unitary similarity and (b) <strong>algebraic</strong> via reduction of $2n \times 2n$ symmetric linearizations of the system by either <em>congruence</em> (Section 4) or <em>similarity</em> (Sections 5 and 6) transformations which are independent of the parameter $\lambda$. Some new results are brought to light in the process. Complete descriptions of associated canonical structures (over $\mathbb{R}$ and over $\mathbb{C}$) are provided -- including the two cases of real symmetric coefficients and complex Hermitian coefficients. These canonical structures include the so-called <em>sign characteristic</em>. This notion appears in the literature with different meanings depending on the choice of canonical form. These sign characteristics are studied here and connections between them are clarified. In particular, we consider which of the linearizations reproduce the (intrinsic) signs associated with the analytic (Rellich) theory (Sections 7 and 9).</p> Peter Lancaster Ion Zaballa Copyright (c) 2021-03-19 2021-03-19 37 211 246 10.13001/ela.2021.5361 Nonsparse companion Hessenberg matrices https://journals.uwyo.edu/index.php/ela/article/view/5405 <p>In recent years, there has been a growing interest in companion matrices. Sparse companion matrices are well known: every sparse companion matrix is equivalent to a Hessenberg matrix of a particular simple type. Recently, Deaett et al. [Electron. J. Linear Algebra, 35:223--247, 2019] started the systematic study of nonsparse companion matrices. They proved that every nonsparse companion matrix is nonderogatory, although not necessarily equivalent to a Hessenberg matrix. In this paper, the nonsparse companion matrices which are unit Hessenberg are described. In a companion matrix, the variables are the coordinates of the characteristic polynomial with respect to the monomial basis. A PB-companion matrix is a generalization, in the sense that the variables are the coordinates of the characteristic polynomial with respect to a general polynomial basis. The literature provides examples with Newton basis, Chebyshev basis, and other general orthogonal bases. Here, the PB-companion matrices which are unit Hessenberg are also described.</p> Alberto Borobia Roberto Canogar Copyright (c) 2021-03-05 2021-03-05 37 193 210 10.13001/ela.2021.5405 Nullities for a class of 0-1 symmetric Toeplitz band matrices https://journals.uwyo.edu/index.php/ela/article/view/5705 <p>Let $S(n,k)$ denote the $n \times n$ symmetric Toeplitz band matrix whose first $k$ superdiagonals and first $k$ subdiagonals have all entries $1$, and whose remaining entries are all $0$. For all $n &gt; k &gt;0$ with $k$ even, we give formulas for the nullity of $S(n,k)$. As an application, it is shown that over half of these matrices $S(n,k)$ are nonsingular. For the purpose of rapid computation, we devise an algorithm that quickly computes the nullity of $S(n,k)$ even for extremely large values of $n$ and $k$, when $k$ is even. The algorithm is based on a connection between the nullspace vectors of $S(n,k)$ and the cycles in a certain directed graph.</p> Ron Evans John Greene Mark Van Veen Copyright (c) 2021-03-05 2021-03-05 37 177 192 10.13001/ela.2021.5705