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 On the tensor rank of the 3 x 3 permanent and determinant <p>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.</p> Siddharth Krishna Visu Makam Copyright (c) 2021-06-10 2021-06-10 37 425 433 10.13001/ela.2021.5107 LIGHTS OUT! on graph products over the ring of integers modulo k <p><em>LIGHTS OUT!</em> is a game played on a finite, simple graph. The vertices of the graph are the lights, which may be on or off, and the edges of the graph determine how neighboring vertices turn on or off when a vertex is pressed. Given an initial configuration of vertices that are on, the object of the game is to turn all the lights out. The traditional game is played over $\mathbb{Z}_2$, where the vertices are either lit or unlit, but the game can be generalized to $\mathbb{Z}_k$, where the lights have different colors. Previously, the game was investigated on Cartesian product graphs over $\mathbb{Z}_2$. We extend this work to $\mathbb{Z}_k$ and investigate two other fundamental graph products, the direct (or tensor) product and the strong product. We provide conditions for which the direct product graph and the strong product graph are solvable based on the factor graphs, and we do so using both open and closed neighborhood switching over $\mathbb{Z}_k$.</p> Ryan Munter Travis Peters Copyright (c) 2021 Ryan Munter, Travis Peters 2021-06-10 2021-06-10 37 416 424 10.13001/ela.2021.5601 Fast verification for the Perron pair of an irreducible nonnegative matrix <p>Fast algorithms are proposed for calculating error bounds for a numerically computed Perron root and vector of an irreducible nonnegative matrix. Emphasis is put on the computational efficiency of these algorithms. Error bounds for the root and vector are based on the Collatz--Wielandt theorem, and estimating a solution of a linear system whose coefficient matrix is an $M$-matrix, respectively. We introduce a technique for obtaining better error bounds. Numerical results show properties of the algorithms.</p> Shinya Miyajima Copyright (c) 2021 Shinya Miyajima 2021-05-31 2021-05-31 37 402 415 10.13001/ela.2021.5181 On the density of semisimple matrices in indefinite scalar product spaces <p>For an indefinite scalar product $[x,y]_B = x^HBy$ for $B= \pm B^H \in \mathbf{Gl}_n(\mathbb{C})$ on $\mathbb{C}^n \times \mathbb{C}^n$, it is shown that the set of diagonalizable matrices is dense in the set of all $B$-normal matrices. The analogous statement is also proven for the sets of $B$-selfadjoint, $B$-skewadjoint and $B$-unitary matrices.</p> Ralph John De la Cruz Philip Saltenberger Copyright (c) 2021 Ralph John De la Cruz, Philip Saltenberger 2021-05-17 2021-05-17 37 387 401 10.13001/ela.2021.5509 Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix sequences <p>In recent years,&nbsp; motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that $f$ belongs to $L^1([-\pi,\pi])$ and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence $\{Y_nT_n[f]\}_n$ has been identified, where $n$ is the matrix size, $Y_n$ is the anti-identity matrix, and $T_n[f]$ is the Toeplitz matrix generated by $f$. In this note, the authors consider the multilevel Toeplitz matrix $T_{\bf n}[f]$ generated by $f\in L^1([-\pi,\pi]^k)$, $\bf n$ being a multi-index identifying the matrix-size, and they prove spectral and singular value distribution results for the matrix-sequence $\{Y_{\bf n}T_{\bf n}[f]\}_{\bf n}$ with $Y_{\bf n}$ being the corresponding tensorization of the anti-identity matrix.</p> Paola Ferrari Isabella Furci Stefano Serra-Capizzano Copyright (c) 2021-05-17 2021-05-17 37 370 386 10.13001/ela.2021.5775