Linear Algebra and its Applications最新文献

In recent years, higher-order trace formulas of operator functions have attracted considerable attention to a large part of the perturbation theory community. In this direction, we prove estimates for traces of higher-order derivatives of multivariate operator functions with associated scalar functions arising from multivariate analytic function space and, as a consequence, derive higher-order spectral shift measures for pairs of tuples of commuting contractions under Hilbert-Schmidt perturbations. These results substantially extend the main results of [26], where the estimates were proved for traces of first and second-order derivatives of multivariate operator functions. In the context of the existence of higher-order spectral shift measures, our results extend the relative results of [6], [20] from a single-variable to a multivariate case under Hilbert-Schmidt perturbations. Our results rely crucially on heavy uses of explicit expressions of higher-order derivatives of operator functions and estimates of the divided difference of multivariate analytic functions, which are developed in this paper, along with the spectral theorem of tuple of commuting normal operators. In conclusion, we explore the significance of our results and provide relevant examples.

In this paper, we give the rank of the walk matrix of the Dynkin graph $A_n$, and prove that its Smith normal form is$\text{diag}(\underset{\lceil \frac{n}{2}\rceil }{\underbrace{1,\dots ,1}},0,\dots ,0).$

A graph $G=(V_G,E_G)$ is said to be a cograph if the path $P_4$ does not appear among its induced subgraphs. The vicinal preorder ≺ on the vertex set $V_G$ is defined in terms of inclusions between neighborhoods. The minimum number $\nabla \left(G\right)$ of ≺-chains required to cover G is called the Dilworth number of G. In this paper it is proved that for a cograph G, the multiplicity of every Seidel eigenvalue $\lambda \ne \pm 1$ does not exceed $\nabla \left(G\right)$. This bound turns out to be tight and can be further improved for threshold graphs. Moreover, it is shown that cographs with at least two vertices have no Seidel eigenvalues in the interval $(-1,1)$.

An automorphism u of a vector space is called unipotent of index 2 whenever ${(u-\mathrm{id})}^2=0$. Let b be a non-degenerate symmetric or skewsymmetric bilinear form on a vector space V over a field $F$ of characteristic different from 2.

Here, we characterize the elements of the isometry group of b that are the product of two unipotent isometries of index 2. In particular, if b is skewsymmetric and nondegenerate we prove that an element of the symplectic group of b is the product of two unipotent isometries of index 2 if and only if it has no Jordan cell of odd size for the eigenvalue −1. As an application, we prove that every element of a symplectic group is the product of three unipotent elements of index 2 (and no fewer in general).

For orthogonal groups, the classification closely matches the classification of sums of two square-zero skewselfadjoint operators that was obtained in a recent article [7].

To apportion a complex matrix means to apply a similarity so that all entries of the resulting matrix have the same magnitude. We initiate the study of apportionment, both by unitary matrix similarity and general matrix similarity. There are connections between apportionment and classical graph decomposition problems, including graceful labellings of graphs, Hadamard matrices, and equiangular lines, and potential applications to instantaneous uniform mixing in quantum walks. The connection between apportionment and graceful labellings allows the construction of apportionable matrices from trees. A generalization of the well-known Eigenvalue Interlacing Inequalities using graceful labellings is also presented. It is shown that every rank one matrix can be apportioned by a unitary similarity, but there are $2\times 2$ matrices that cannot be apportioned. A necessary condition for a matrix to be apportioned by unitary matrix is established. This condition is used to construct a set of matrices with nonzero Lebesgue measure that are not apportionable by a unitary matrix.

We present a framework to decompose real multivariate polynomials while preserving invariance and positivity. This framework has been recently introduced for tensor decompositions, in particular for quantum many-body systems. Here we transfer results about decomposition structures, invariance under permutations of variables, positivity, rank inequalities and separations, approximations, and undecidability to real polynomials. Specifically, we define invariant decompositions of polynomials, and characterize which polynomials admit such decompositions. We then include positivity: We define invariant separable and sum-of-squares decompositions, and characterize the polynomials similarly. We provide inequalities and separations between the ranks of the decompositions, and show that the separations are not robust with respect to approximations. For cyclically invariant decompositions, we show that it is undecidable whether the polynomial is nonnegative or sum-of-squares for all system sizes. Our framework is different from existing approaches for polynomial decompositions, since it covers symmetry and positivity combined, in a clean and uniform way. Also, our work sheds new light on polynomials by putting them on an equal footing with tensors, and opens the door to extending this framework to other tensor product structures.

In this paper, we provide a formula to compute the permanent of block matrices depending on entries of each block. As a consequence, a generalized Lieb permanent inequality on positive semi-definite block matrices is given.