Linear Algebra and its Applications最新文献

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.

We show that the Heisenberg Lie algebras over a field $F$ of characteristic $p>0$ admit a family of restricted Lie algebras, and we classify all such non-isomorphic restricted Lie algebra structures. We use the ordinary 1- and 2-cohomology spaces with trivial coefficients to compute the restricted 1- and 2-cohomology spaces of these restricted Heisenberg Lie algebras. We describe the restricted 1-dimensional central extensions, including explicit formulas for the Lie brackets and ${\cdot }^{\left[p\right]}$-operators.

In this paper we look at the families of random walks arising from FI-graphs. One may think of these objects as families of nested graphs, each equipped with a natural action by a symmetric group $S_n$, such that these actions are compatible and transitive. Families of graphs of this form were introduced by the authors in [9], while a systematic study of random walks on these families were considered in [10]. In the present work, we illustrate that these random walks never exhibit the so-called product condition, and therefore also never display total variation cutoff as defined by Aldous and Diaconis [1]. In particular, we provide a large family of algebro-combinatorially motivated examples of collections of Markov chains which satisfy some well-known algebraic heuristics for cutoff, while not actually having the property.

It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results such as Ihara's theorem and the zeta function on graphs. In Grindrod et al. [13], the radius of convergence of the generating function was studied for simple (i.e., undirected, unweighted and with no loops) graphs, and shown to depend on the number of cycles in the graph. In this paper, we use technologies from the theory of polynomial and rational matrices to greatly extend these results by studying the radius of convergence of the corresponding generating function for general, possibly directed and/or weighted, graphs. We give an analogous characterization of the radius of convergence for directed (unweighted or weighted) graphs, showing that it depends on the number of cycles in the undirectization of the graph. We also consider backtrack-downweighted walks on unweighted digraphs, and we prove a version of Ihara's theorem in that case. Finally, for weighted directed graphs, we provide for the first time an exact formula for the radius of convergence, improving a previous result that exhibited a lower bound, and we also prove a version of Ihara's theorem.

Let G be a simple connected graph and $A\left(G\right)$ be the adjacency matrix of G. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If $A\left(G\right)$ is nonsingular and $X=SA{\left(G\right)}^{-1}{S}^{-1}$ is entrywise nonnegative for some signature matrix S, then X can be viewed as the adjacency matrix of a unique weighted graph. It is called the inverse of G, denoted by $G^{+}$. A graph G is said to have the reciprocal eigenvalue property (property(R)) if $A\left(G\right)$ is nonsingular, and $\frac{1}{\lambda }$ is an eigenvalue of $A\left(G\right)$ whenever λ is an eigenvalue of $A\left(G\right)$. Further, if λ and $\frac{1}{\lambda }$ have the same multiplicity for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (property (SR)). It is known that for a tree T, the following conditions are equivalent: a) $T^{+}$ is isomorphic to T, b) T has property (R), c) T has property (SR) and d) T is a corona tree (it is a tree which is obtained from another tree by adding a new pendant at each vertex).

Studies on the inverses, property (R) and property (SR) of bipartite graphs are available in the literature. However, their studies for the non-bipartite graphs are rarely done. In this article, we study the inverse and property (SR) for non-bipartite graphs. We first introduce an operation, which helps us to study the inverses of non-bipartite graphs. As a consequence, we supply a class of non-bipartite graphs for which the inverse graph $G^{+}$ exists and $G^{+}$ is isomorphic to G. It follows that each graph G in this class has property (SR).

Let F be a field and $q\left(x\right)$ a quadratic polynomial in $F\left[x\right]$ with $q\left(0\right)\ne 0$. We denote by $T_{\infty }\left(F\right)$ the algebra of all infinite upper triangular matrices over the field F. A matrix $A\in T_{\infty }\left(F\right)$ is called a quadratic matrix with respect to $q\left(x\right)$ if $q\left(A\right)=0$. In this paper, we first investigate the subgroup in $T_{\infty }\left(F\right)$ generated by all quadratic matrices with respect to $q\left(x\right)$ and then present some applications.