The Electronic Journal of Linear Algebra

The numerical range of matrix products

Stephen Drury — Mon, 04 Mar 2024 00:00:00 -0700

We discuss what can be said about the numerical range of the matrix product $A_1A_2$ when the numerical ranges of $A_1$ and $A_2$ are known. If two compact convex subsets $K_1, K_2$ of the complex plane are given, we discuss the issue of finding a compact convex subset $K$ such that whenever $A_j$ ($j=1,2$) are either unrestricted matrices or normal matrices of the same shape with $W(A_j) \subseteq K_j$, it follows that $W(A_1A_2) \subseteq K$. We do this by defining specific deviation quantities for both the unrestricted case and the normal case.

Diagonalizably realizable implies universally realizable

Carlos Marijuán, Ricardo L. Soto — Tue, 23 Apr 2024 00:00:00 -0700

A spectrum $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix $A$. The spectrum $\Lambda$ is diagonalizably realizable ($\mathcal{DR}$) if the realizing matrix $A$ is diagonalizable, and $\Lambda$ is universally realizable ($\mathcal{UR}$) if it is realizable for each possible Jordan canonical form allowed by $\Lambda.$ In 1981, Minc proved that if $\Lambda$ is the spectrum of a diagonalizable positive matrix, then $\Lambda$ is universally realizable. One of the main open questions about the problem of universal realizability of spectra is
whether $\mathcal{DR}$ implies $\mathcal{UR}$. Here, we prove a surprisingly simple result, which shows how diagonalizably realizable implies universally realizable.

Commuting additive maps on upper triangular and strictly upper triangular infinite matrices

Di-Chen Lan, Cheng-Kai Liu — Thu, 11 Apr 2024 00:00:00 -0700

Let ${\mathbb F}$ be a field, let $N_{\infty}({\mathbb F})$ be the ring of all ${\mathbb N}\times {\mathbb N}$ strictly upper triangular matrices over ${\mathbb F,}$ and let $T_{\infty}({\mathbb F})$ be the ring of all ${\mathbb N}\times {\mathbb N}$ upper triangular matrices over ${\mathbb F}$. In this paper, we completely characterize additive maps $f:N_{\infty}({\mathbb F})\to T_{\infty}({\mathbb F})$ satisfying $[f(x),x]=0$ for all $x\in N_{\infty}({\mathbb F})$. As applications, we obtain the finite fields versions of the two main results recently obtained by Slowik and Ahmed [Electron. J. Linear Algebra 37:247-255, 2021].

A new weighted spectral geometric mean and properties

Trung Hoa Dinh, Tin-Yau Tam, Trung-Dung Vuong — Wed, 06 Mar 2024 00:00:00 -0700

In this paper, we introduce a new weighted spectral geometric mean: \begin{equation*}\label{F-mean}
F_t(A,B)= (A^{-1}\sharp_t B)^{1/2} A^{2-2t} (A^{-1} \sharp_t B)^{1/2}, \quad t\in [0,1],
\end{equation*} where $A$ and $B$ are positive definite matrices. We study basic properties and inequalities for $F_t(A, B)$. We also establish the Lie-Trotter formula for $F_t(A, B)$. Finally, we extend some of the results on $F_t(A, B)$ to symmetric space of noncompact types.

Some symmetric sign patterns requiring unique inertia

Partha Rana, Sriparna Bandopadhyay — Tue, 23 Apr 2024 00:00:00 -0700

A sign pattern is a matrix whose entries are from the set $\{+,-,0\}$. A sign pattern requires unique inertia if every matrix in its qualitative class has the same inertia. For symmetric tree sign patterns, several necessary and sufficient conditions to require unique inertia are known. In this paper, sufficient conditions for symmetric tree sign patterns to require unique inertia based on the sign and position of the loops in the underlying graph are given. Further, some sufficient conditions for a symmetric sign pattern to require unique inertia if the underlying graph contains cycles are determined.