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 Trees with maximum sum of the two largest Laplacian eigenvalues <p>Let $T$ be a tree of order $n$ and $S_2(T)$ be the sum of the two largest Laplacian eigenvalues of $T$. Fritscher <em>et al.</em> proved that for any tree $T$ of order $n$, $S_2(T) \leq n+2-\frac{2}{n}$. Guan <em>et al.</em> determined the tree with maximum $S_2(T)$ among all trees of order $n$. In this paper, we characterize the trees with $S_2(T) \geq n+1$ among all trees of order $n$ except some trees. Moreover, among all trees of order $n$, we also determine the first $\lfloor\frac{n-2}{2}\rfloor$ trees according to their $S_2(T)$. This extends the result of Guan <em>et al.</em></p> Yirong Zheng Jianxi Li Sarula Chang Copyright (c) 2022 Yirong Zheng, Jianxi Li, Sarula Chang 2022-07-15 2022-07-15 357 366 10.13001/ela.2022.7065 Positive linear maps and spreads of normal matrices <p>We obtain some inequalities involving positive linear maps on matrix algebra. The special cases provide bounds for the spreads of normal matrices.</p> Rajesh Sharma Manish Pal Copyright (c) 2022 Rajesh Sharma, Manish Pal 2022-06-21 2022-06-21 347 356 10.13001/ela.2022.7009 K-subdirect sums of Nekrasov matrices <p>In this paper, we give a sufficient and necessary condition for the subdirect sum of a Nekrasov matrix and a strictly diagonally dominant matrix being still a Nekrasov matrix. Adopting this sufficient and necessary condition, we present several simple sufficient conditions ensuring that the subdirect sum of Nekrasov matrices is in the same class. Examples are reported to illustrate the theoretical results.</p> Zhen-Hua Lyu Xueru Wang Lishu Wen Copyright (c) 2022 Zhen-Hua Lyu, Xueru Wang, Lishu Wen 2022-06-10 2022-06-10 339 346 10.13001/ela.2022.6951 Linear maps preserving the Lorentz spectrum: the $2 \times 2$ case <p>In this paper, a complete description of the linear maps $\phi:W_{n}\rightarrow W_{n}$ that preserve the Lorentz spectrum is given when $n=2$, and $W_{n}$ is the space $M_{n}$ of $n\times n$ real matrices or the subspace $S_{n}$ of $M_{n}$ formed by the symmetric matrices. In both cases, it has been shown that $\phi(A)=PAP^{-1}$ for all $A\in W_{2}$, where $P$ is a matrix with a certain structure. It was also shown that such preservers do not change the nature of the Lorentz eigenvalues (that is, the fact that they are associated with Lorentz eigenvectors in the interior or on the boundary of the Lorentz cone). These results extend to $n=2$ those for $n\geq 3$ obtained by Bueno, Furtado, and Sivakumar (2021). The case $n=2$ has some specificities, when compared to the case $n\geq3,$ due to the fact that the Lorentz cone in $\mathbb{R}^{2}$ is polyedral, contrary to what happens when it is contained in $\mathbb{R}^{n}$ with $n\geq3.$ Thus, the study of the Lorentz spectrum preservers on $W_n = M_n$ also follows from the known description of the Pareto spectrum preservers on $M_n$.</p> MarĂ­a I. Bueno Susana Furtado Aelita Klausmeier Joey Veltri Copyright (c) 2022 M. I. Bueno, Susana Furtado, Aelita Klausmeier, Joey Veltri 2022-06-02 2022-06-02 317 330 10.13001/ela.2022.6925 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