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 Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem <p>Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, have been extended to the case of symplectic eigenvalues. In this note, we will generalize Williamson's diagonal form for symmetric positive-definite matrices to the case of symmetric positive-semidefinite matrices, which allows us to define symplectic eigenvalues, and prove the trace minimization theorem in the new setting.</p> Nguyen Thanh Son Tatjana Stykel Copyright (c) 2022 Dr. Nguyen Thanh Son, Prof. Dr. Tatjana Stykel 2022-10-06 2022-10-06 607 616 10.13001/ela.2022.7351 Recovering the characteristic polynomial of a graph from entries of the adjugate matrix <p>The adjugate matrix of $G$, denoted by $\operatorname{adj}(G)$, is the adjugate of the matrix $x\mathbf{I}-\mathbf{A}$, where $\mathbf{A}$ is the adjacency matrix of $G$. The polynomial reconstruction problem (PRP) asks if the characteristic polynomial of a graph $G$ can always be recovered from the multiset $\operatorname{\mathcal{PD}}(G)$ containing the $n$ characteristic polynomials of the vertex-deleted subgraphs of $G$. Noting that the $n$ diagonal entries of $\operatorname{adj}(G)$ are precisely the elements of $\operatorname{\mathcal{PD}}(G)$, we investigate variants of the PRP in which multisets containing entries from $\operatorname{adj}(G)$ successfully reconstruct the characteristic polynomial of $G$. Furthermore, we interpret the entries off the diagonal of $\operatorname{adj}(G)$ in terms of characteristic polynomials of graphs, allowing us to solve versions of the PRP that utilize alternative multisets to $\operatorname{\mathcal{PD}}(G)$ containing polynomials related to characteristic polynomials of graphs, rather than entries from $\operatorname{adj}(G)$.</p> Alexander Farrugia Copyright (c) 2022 Alexander Farrugia 2022-10-28 2022-10-28 697 711 10.13001/ela.2022.7231 New results on $M$-matrices, $H$-matrices and their inverse classes <p>In this article, some new results on $M$-matrices, $H$-matrices and their inverse classes are proved. Specifically, we study when a singular $Z$-matrix is an $M$-matrix, convex combinations of $H$-matrices, almost monotone $H$-matrices and Cholesky factorizations of $H$-matrices.</p> Samir Mondal K. C. Sivakumar Michael Tsatsomeros Copyright (c) 2022 Samir Mondal, K. C. Sivakumar, Michael Tsatsomeros 2022-11-23 2022-11-23 729 744 10.13001/ela.2022.7177 Sums of orthogonal, symmetric, and skew-symmetric matrices <p>An $n$-by-$n$ matrix $A$ is called symmetric, skew-symmetric, and orthogonal if $A^T=A$, $A^T=-A$, and $A^T=A^{-1}$, respectively. We give necessary and sufficient conditions on a complex matrix $A$ so that it is a sum of type ``"orthogonal $+$ symmetric" in terms of the Jordan form of $A-A^T$. We also give necessary and sufficient conditions on a complex matrix $A$ so that it is a sum of type "orthogonal $+$ skew-symmetric" in terms of the Jordan form of $A+A^T$.</p> Ralph John de la Cruz Agnes T. Paras Copyright (c) 2022 Ralph John de la Cruz, Agnes T. Paras 2022-10-07 2022-10-07 655 660 10.13001/ela.2022.7129 A note on bounds for eigenvalues of nonsingular H-tensors <p>A counterexample to a theorem in the paper ELA 29:3-16, (2015) is provided, and an upper bound on the H-spectral radius of H-tensors is given.</p> Jun He Guanjun Xu Copyright (c) 2022 Jun He, Guanjun Xu 2022-08-20 2022-08-20 483 485 10.13001/ela.2022.7097