The Electronic Journal of Linear Algebra

Periodic two-dimensional descriptor systems

Peter Benner — 2023-08-24

In this note, we analyze the compatibility conditions of 2D descriptor systems with periodic coefficients and we derive a special coordinate system in which these conditions reduce to simple matrix commutativity conditions. We also show that the compatibility of the different trajectories in such a periodic 2D descriptor system can elegantly be formulated in terms of so-called matrix relations of regular pencils, which were introduced in [Benner and Byers. An arithmetic for matrix pencils: Theory and new algorithms. Numer. Math., 103(4):539-573, 2006]. We then show that these ideas can be extended to multidimensional periodic descriptor systems and briefly discuss the difference between the case of complex and real coefficient matrices.

The inverse of a symmetric nonnegative matrix can be copositive

Robert Reams — 2023-09-27

Let $A$ be an $n\times n$ symmetric matrix. We first show that if $A$ and its pseudoinverse are strictly copositive, then $A$ is positive semidefinite, which extends a similar result of Han and Mangasarian. Suppose $A$ is invertible, as well as being symmetric. We showed in an earlier paper that if $A^{-1}$ is nonnegative with $n$ zero diagonal entries, then $A$ can be copositive (for instance, this happens with the Horn matrix), and when $A$ is copositive, it cannot be of form $P+N$, where $P$ is positive semidefinite and $N$ is nonnegative and symmetric. Here, we show that if $A^{-1}$ is nonnegative with $n-1$ zero diagonal entries and one positive diagonal entry, then $A$ can be of the form $P+N$, and we show how to construct $A$. We also show that if $A^{-1}$ is nonnegative with one zero diagonal entry and $n-1$ positive diagonal entries, then $A$ cannot be copositive.

Decompositions of matrices into a sum of invertible matrices and matrices of fixed nilpotence

Peter Danchev — 2023-08-24

For any $n\ge 2$ and fixed $k\ge 1$, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring $\mathbb{M}_n(\mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix $N$ with $N^k=0$ over an arbitrary field $\mathbb{F}$.

Pareto H-eigenvalues of nonnegative tensors and uniform hypergraphs

Lu Zheng — 2023-08-02

The Pareto H-eigenvalues of nonnegative tensors and (adjacency tensors of) uniform hypergraphs are studied. Particularly, it is shown that the Pareto H-eigenvalues of a nonnegative tensor are just the spectral radii of its weakly irreducible principal subtensors, and those hypergraphs that minimize or maximize the second largest Pareto H-eigenvalue over several well-known classes of uniform hypergraphs are determined.

Positive and negative square energies of graphs

Aida Abiad — 2023-06-08

The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Let $s^+(G), s^-(G)$ denote the sum of the squares of the positive and negative eigenvalues of $G$, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if $G$ is a connected graph of order $n$, then $s^+(G)\geq n-1$ and $s^-(G) \geq n-1$. In this paper, we show partial results towards this conjecture. In particular, numerous structural results that may help in proving the conjecture are derived, including the effect of various graph operations. These are then used to establish the conjecture for several graph classes, including graphs with certain fraction of positive eigenvalues and unicyclic graphs.