Linear Algebra and its Applications最新文献

We prove the inequality ${\lambda }_k\le d_k+d_{k-1}$ for all the eigenvalues ${\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_n$ of the Kirchhoff matrix K of a finite simple graph or quiver with vertex degrees $d_1\le d_2\le \cdots \le d_n$ and assuming $d_0=0$. Without multiple connections, the inequality ${\lambda }_k\ge \max (0,d_k-(n-k\left)\right)$ holds. A consequence in the finite simple graph or multi-graph case is that the pseudo determinant $\mathrm{Det}\left(K\right)$ counting the number of rooted spanning trees has an upper bound $2^n{\prod }_{k=1}^n{d}_k$ and that $\det (1+K)$ counting the number of rooted spanning forests has an upper bound ${\prod }_{k=1}^n(1+2d_k)$.

The main purpose of this paper is to study the class of Lie-admissible algebras $(A,.)$ such that its product is a biderivation of the Lie algebra $(A,[,\left]\right)$, where $[,]$ is the commutator of the algebra $(A,.)$. First, we provide characterizations of algebras in this class. Furthermore, we show that this class of nonassociative algebras includes Lie algebras, symmetric Leibniz algebras, Lie-admissible left (or right) Leibniz algebras, Milnor algebras, and LR-algebras. Then, we establish results on the structure of these algebras in the case that the underlying Lie algebras are perfect (in particular, semisimple Lie algebras). In addition, we then study flexible $A_{BD}$-algebras, showing in particular that these algebras are extensions of Lie algebras in the category of flexible $A_{BD}$-algebras. Finally, we study left-symmetric $A_{BD}$-algebras, in particular we are interested in flat pseudo-Euclidean Lie algebras where the associated Levi-Civita products define $A_{BD}$-algebras on the underlying vector spaces of these Lie algebras. In addition, we obtain an inductive description of all these Lie algebras and their Levi-Civita products (in particular, for all signatures in the case of real Lie algebras).

A unified framework for the Expander Mixing Lemma for irregular graphs using adjacency eigenvalues is presented, as well as two new versions of it. While the existing Expander Mixing Lemmas for irregular graphs make use of the notion of volume (the sum of degrees within a vertex set), we instead propose to use the Perron eigenvector entries as vertex weights, which is a way to regularize the graph. This provides a new application of weight partitions of graphs. The new Expander Mixing Lemma versions are then applied to obtain several eigenvalue bounds for NP-hard parameters such as the zero forcing number, the vertex integrity and the routing number of a graph.

Huang, McKinnon, and Satriano conjectured that if $v\in R^n$ has distinct coordinates and $n\ge 3$, then a hyperplane through the origin other than ${\sum }_i{x}_i=0$ contains at most $2\lfloor n/2\rfloor (n-2)!$ of the vectors obtained by permuting the coordinates of v. We prove this conjecture.

In this work we consider generic losses of rank for complex valued matrix functions depending on two parameters. We give theoretical results that characterize parameter regions where these losses of rank occur. Our main results consist in showing how following an appropriate smooth SVD along a closed loop it is possible to monitor the Berry phases accrued by the singular vectors to decide if –inside the loop– there are parameter values where a loss of rank takes place. It will be needed to use a new construction of a smooth SVD, which we call the “joint-MVD” (minimum variation decomposition).

We establish novel two-sided bounds for the tracial seminorm of multilinear Schur multipliers that tighten previously known bounds. The result is obtained by a newly developed method based on polynomial chaoses.

A linear group is called unisingular if every element of it has eigenvalue 1. In this paper we develop some general machinery for the study of unisingular irreducible linear groups. A motivation for the study of such groups comes from several sources, including algebraic geometry, Galois theory, finite group theory and representation theory. In particular, a certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements, and in this paper we concentrate on this special case of the general problem. A more special but important question is that of the existence of such subgroups in the symplectic groups of particular degrees. We answer this question for almost all degrees $2n<250$, specifically, the question remains open only 7 values of n.

We construct and investigate certain (unbalanced) superalgebra structures on ${\text{End}}_K\left(V\right)$, with K a field of characteristic 0 and V a finite dimensional K-vector space (of dimension $n\ge 2$). These structures are induced by a choice of non-degenerate symmetric bilinear form B on V and a choice of non-zero base vector $w\in V$. After exploring the construction further, we apply our results to certain questions concerning integer matrix factorization and isometry of integral lattices.

Optimizing strongly convex functions subject to linear constraints is a fundamental problem with numerous applications. In this work, we propose a block (accelerated) randomized Bregman-Kaczmarz method that only uses a block of constraints in each iteration to tackle this problem. We consider a dual formulation of this problem in order to deal in an efficient way with the linear constraints. Using convex tools, we show that the corresponding dual function satisfies the Polyak-Lojasiewicz (PL) property, provided that the primal objective function is strongly convex and verifies additionally some other mild assumptions. However, adapting the existing theory on coordinate descent methods to our dual formulation can only give us sublinear convergence results in the dual space. In order to obtain convergence results in some criterion corresponding to the primal (original) problem, we transfer our algorithm to the primal space, which combined with the PL property allows us to get linear convergence rates. More specifically, we provide a theoretical analysis of the convergence of our proposed method under different assumptions on the objective and demonstrate in the numerical experiments its superior efficiency and speed up compared to existing methods for the same problem.

A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra $g$ is splittable if and only if $g=a+n$, where $a$ is an abelian subalgebra of $g$ composed of semisimple elements and $n$ is the ideal of all nilpotent matrices of $g$. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements.