In this paper we studied the classical numerical range of matrices in sp(2n, C). We obtained some result on the relationship between the numerical range of a matrix in and that [...] of its diagonal block, the singular values of its off-diagonal block A2.

A matrix $𝒜$ whose entries come from the set $\{+,-,0\}$ is called a sign pattern matrix, or sign pattern. A sign pattern is said to be potentially nilpotent if it has a nilpotent realization. In this paper, the characterization problem for some potentially nilpotent double star sign patterns is discussed. A class of double star sign patterns, denoted by $𝒟SSP(m,2)$, is introduced. We determine all potentially nilpotent sign patterns in $𝒟SSP(3,2)$ and $𝒟SSP(5,2)$, and prove that one sign pattern in $𝒟SSP(3,2)$ is potentially stable.

Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix [...] there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.

If $\left\{v_i\right\}$ and $\left\{w_j\right\}$ are two families of unitary bases for ${\mathbf{C}}^n$, and $\theta$ is a fixed number, let $V^n$ and $W^n$ be subspaces of ${\mathbf{C}}^n$ spanned by $[\theta ·n]$ vectors in $\left\{v_i\right\}$ and $\left\{w_j\right\}$ respectively. We study the angle between $V^n$ and $W^n$ as $n$ goes to infinity. We show that when $\left\{v_i\right\}$ and $\left\{w_j\right\}$ arise in certain arithmetically defined families, the angles between $V^n$ and $W^n$ may either tend to $0$ or be bounded away from zero, depending on the behavior of an associated eigenvalue problem.

Let Ω be the spectral unit ball of Mₙ(ℂ), that is, the set of n × n matrices with spectral radius less than 1. We are interested in classifying the automorphisms of Ω. We know that it is enough to consider the normalized automorphisms of Ω, that is, the automorphisms F satisfying F(0) = 0 and F'(0) = I, where I is the identity map on Mₙ(ℂ). The known normalized automorphisms are conjugations. Is every normalized automorphism a conjugation? We show that locally, in a neighborhood of a matrix with...

Let ${\mu }_{n-1}\left(G\right)$ be the algebraic connectivity, and let ${\mu }_1\left(G\right)$ be the Laplacian spectral radius of a $k$-connected graph $G$ with $n$ vertices and $m$ edges. In this paper, we prove that $${\mu }_{n-1}\left(G\right)\ge \frac{2n{k}^2}{(n(n-1)-2m)(n+k-2)+2k^2},$$ with equality if and only if $G$ is the complete graph $K_n$ or $K_n-e$. Moreover, if $G$ is non-regular, then $${\mu }_1\left(G\right)<2\Delta -\frac{2(n\Delta -2m)k^2}{2(n\Delta -2m)(n^2-2n+2k)+n{k}^2},$$ where $\Delta$ stands for the maximum degree of $G$. Remark that in some cases, these two inequalities improve some previously known results.