Linear Algebra and its Applications最新文献

By employing a weighted Frobenius norm with a positive definite matrix ω, we introduce natural generalizations of the famous Böttcher-Wenzel (BW) inequality. Based on the combination of the weighted Frobenius norm

Given a pair of real symmetric matrices $A,B\in R^{n\times n}$ with nonzero patterns determined by the edges of any pair of chosen graphs on n vertices, we consider an inverse eigenvalue problem for the structured matrix $C=\left[\begin{array}{cc}A& B\\ I& O\end{array}\right]\in R^{2n\times 2n}$. We conjecture that C can attain any spectrum that is closed under conjugation. We use a structured Jacobian method to prove this conjecture for A and B of orders at most 4 or when the graph of A has a Hamilton path, and prove a weaker version of this conjecture for any pair of graphs with a restriction on the multiplicities of eigenvalues of C.

A complex unit gain graph, or $T$-gain graph, is a triple $\Phi =(G,T,\phi )$ comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers $T=\{z\in C:|z|=1\}$, and a gain function $\phi :\overrightarrow{E}\to T$ with the property that $\phi \left(e_{ij}\right)=\phi {\left(e_{ji}\right)}^{-1}$. A cactus graph is a connected graph in which any two cycles have at most one vertex in common.

In this paper, we firstly show that there does not exist a complex unit gain graph with nullity $n\left(G\right)-2m\left(G\right)+2c\left(G\right)-1$, where $n\left(G\right)$, $m\left(G\right)$ and $c\left(G\right)$ are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30].

In this article, taking a Fiedler's result on the spectrum of a matrix formed from two symmetric matrices as a motivation, we deduce a more general result on the eigenvalues of a matrix, which form from n symmetric matrices. As an important application, we obtain the adjacency spectra, Laplacian spectra and signless Laplacian spectra of a graph with a particular almost equitable partition.

We establish necessary and sufficient conditions for invertibility of symmetric three-by-three block matrices having a double saddle-point structure that guarantee the unique solvability of double saddle-point systems. We consider various scenarios, including the case where all diagonal blocks are allowed to be rank deficient. Under certain conditions related to the nullity of the blocks and intersections of their kernels, an explicit formula for the inverse is derived.

In this paper, we solve a classical counting problem for non-degenerate forms of symplectic and hermitian type defined on a vector space: given a subspace π, we find the number of non-singular subspaces that are trivially intersecting with π and span a non-singular subspace with π. Lower bounds for the quantity of such pairs where π is non-singular were first studied in “Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)”, which was later improved in “Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)” and generalised in “Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)”. In this paper, we derive explicit formulae, which allow us to give the exact proportion and improve the known lower bounds.

In this paper, we study the representation of an infinite-dimensional Lie algebra $C$ related to the q-analog Virasoro-like Lie algebra. We give the necessary and sufficient conditions for the highest weight irreducible module $V\left(\varphi \right)$ of $C$ to be a Harish-Chandra module. We prove that the Verma $C$-module $\overline{V}\left(\varphi \right)$ is either irreducible or has the corresponding irreducible highest weight $C$-module $V\left(\varphi \right)$ that is a Harish-Chandra module. We also give the maximal proper submodule of the Verma module $\overline{V}\left(\varphi \right)$ and the e-character of the irreducible highest weight $C$-module $V\left(\varphi \right)$ when the highest weight ϕ satisfies some natural conditions. Furthermore, we give the classification of the Harish-Chandra $C$-modules with nontrivial central charge.

In the Analytic Hierarchy Process (AHP) the efficient vectors for a pairwise comparison matrix (PC matrix) are based on the principle of Pareto optimal decisions. To infer the efficiency of a vector for a PC matrix we construct a directed Hamiltonian cycle of a certain digraph associated with the PC matrix and the vector. We describe advantages of using this process over using the strong connectivity of the digraph. As an application of our process we find efficient vectors for a PC matrix, A, obtained from a consistent one by perturbing three entries above the main diagonal and the corresponding reciprocal entries, in a way that there is a square submatrix of A of order 2 containing three of the perturbed entries and not containing a diagonal entry of A. For completeness, we include examples showing conditions under which, when deleting a certain entry of an efficient vector for the square matrix A of order n, we have a non-efficient vector for the corresponding square principal submatrix of order n-1 of A.

A P-matrix is a matrix all of whose principal minors are positive. We demonstrate that the fractional powers of a P-matrix are also P-matrices. This insight allows us to affirmatively address a longstanding conjecture raised in Hershkowitz and Johnson (1986) [8]: It is shown that if $A^k$ is a P-matrix for all positive integers k, then the eigenvalues of A are positive.