The Electronic Journal of Linear Algebra

The numerical range of matrix products

Stephen Drury — 2024-03-04

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.

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

Di-Chen Lan — 2024-04-11

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 — 2024-03-06

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.

Flag-shaped blockers of 123-avoiding permutation matrices

Megan Bennett — 2024-02-07

A blocker of $123$-avoiding permutation matrices refers to the set of zeros contained within an $n\times n$ $123$-forcing matrix. Recently, Brualdi and Cao provided a characterization of all minimal blockers, which are blockers with a cardinality of $n$. Building upon their work, a new type of blocker, flag-shaped blockers, which can be seen as a generalization of the $L$-shaped blockers defined by Brualdi and Cao, are introduced. It is demonstrated that all flag-shaped blockers are minimum blockers. The possible cardinalities of flag-shaped blockers are also determined, and the dimensions of subpolytopes that are defined by flag-shaped blockers are examined.

New properties of a special matrix related to positive-definite matrices

Shaowu Huang — 2024-01-30

Let $H$ be a $2n\times 2n$ real symmetric positive-definite matrix. Suppose that $H\circ H=(H_{ij})_{2n\times 2n}$ is a partitioned matrix, in which $\circ$ represents the Hadamard product and the block $H_{ij}$ has order $n\times n$, $1\leq i,j \leq 2$. Several new properties on the matrix $\widetilde{H}$ are derived including inequalities that involve the symplectic eigenvalues and the usual eigenvalues, where $2\widetilde{H}=H_{11}+H_{22}+H_{12}+H_{21}$.