Linear Algebra and its Applications最新文献

In this paper, we provide a structure theorem and various characterizations of degradable strong entanglement breaking maps on separable Hilbert spaces. In the finite-dimensional case, we prove that unital degradable entanglement breaking maps are precisely the $C^{⁎}$-extreme points of the convex set of unital entanglement breaking maps on matrix algebras. Consequently, we get a structure for unital degradable positive partial transpose (PPT) maps.

Zero forcing and maximum nullity are two important graph parameters which have been laboriously studied in order to aid in the resolution of the Inverse Eigenvalue problem. Motivated in part by an observation that the zero forcing number for the complement of a tree on n vertices is either $n-3$ or $n-1$ in one exceptional case, we consider the zero forcing number for the complement of more general graphs under certain conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs. Finally, we yield equality between the maximum nullity and zero forcing number of several families of graph complements considered.

For a weighted graph G, the rotation of an edge $u{v}_1$ from $v_1$ to a vertex $v_2$ is defined as follows: delete the edge $u{v}_1$, set $w\left(u{v}_2\right)$ as $w\left(u{v}_1\right)+w\left(u{v}_2\right)$ if $u{v}_2$ is an edge of G; otherwise, add a new edge $u{v}_2$ and set $w\left(u{v}_2\right)=w\left(u{v}_1\right)$, where $w\left(u{v}_1\right)$ and $w\left(u{v}_2\right)$ are the weights of the edges $u{v}_1$ and $u{v}_2$, respectively. In this paper, effects on the algebraic connectivity of weighted graphs under edge rotations are studied. For a weighted graph, a sufficient condition for an edge rotation to reduce its algebraic connectivity and a necessary condition for an edge rotation to improve its algebraic connectivity are proposed based on Fiedler vectors of the graph. As applications, we show that, by using a series of edge rotations, a pair of pendent paths (a pendent tree) of a weighted graph can be transformed into one pendent path (pendent edges attached at a common vertex) of the graph with the algebraic connectivity decreasing (increasing) monotonically. These results extend previous findings of reducing the algebraic connectivity of unweighted graphs by using edge rotations.

In a recent article Projective geometries, Q-polynomial structures, and quantum groups Terwilliger (arXiv:2407.14964) defined a certain weighted adjacency matrix, depending on a free (positive real) parameter, associated with the projective geometry, and showed (among many other results) that it is diagonalizable, with the eigenvalues and their multiplicities explicitly written down, and that it satisfies the Q-polynomial property (with respect to the zero subspace).

In this note we

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  Write down an explicit eigenbasis for this matrix.

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  Evaluate the adjacency matrix-eigenvector products, yielding a new proof for the eigenvalues and their multiplicities.

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  Evaluate the dual adjacency matrix-eigenvector products and directly show that the action of the dual adjacency matrix on the eigenspaces of the adjacency matrix is block-tridiagonal, yielding a new proof of the Q-polynomial property.

Let $F$ be a field with characteristic not equal to 2, and $S\in M_{2n}\left(F\right)$ be skew-symmetric and nonsingular. For $X\in M_{2n}\left(F\right)$, we show that X has a ${\varphi }_S$ polar decomposition if and only if (a) ${\varphi }_S\left(X\right)X$ has a ${\varphi }_S$-symmetric square root, (b) $X{\varphi }_S\left(X\right)$ is similar to ${\varphi }_S\left(X\right)X$, and (c) rank ${\left[X{\varphi }_S\right(X\left)\right]}^{t}X$ is even for all nonnegative integers t.

In this paper we explore the relation between the A-numerical range and the A-spectrum of A-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of A-normal operator and prove that closure of the A-numerical range of an A-normal operator is the convex hull of the A-spectrum. We further prove Anderson's theorem for the sum of A-normal and A-compact operators which improves and generalizes the existing result on Anderson's theorem for A-compact operators. Finally we introduce strongly A-numerically closed class of operators and along with other results prove that the class of A-normal operators is strongly A-numerically closed.

Recent developments in the spectral theory of Bayesian Networks has led to a need for a developed theory of estimation and inference on the eigenvalues of the normalized precision matrix, Ω. In this paper, working under conditions where $n\to \infty$ and p remains fixed, we provide multivariate normal asymptotic distributions of the sample eigenvalues of Ω under general conditions and under normal populations, a formula for second-order bias correction of these sample eigenvalues, and a Stein-type shrinkage estimator of the eigenvalues. Numerical simulations are performed which demonstrate under what generative conditions each estimation technique is most effective. When the largest eigenvalue of Ω is small the simulations show that the second order bias-corrected eigenvalue was considerably less biased than the sample eigenvalue, whereas the smallest eigenvalue was estimated with less bias using either the sample eigenvalue or the proposed shrinkage method.

We consider the inverse problem of finding a magnitude-symmetric matrix (matrix with opposing off-diagonal entries equal in magnitude) with a prescribed set of principal minors. This problem is closely related to the theory of recognizing and learning signed determinantal point processes in machine learning, as kernels of these point processes are magnitude-symmetric matrices. In this work, we prove a number of properties regarding sparse and generic magnitude-symmetric matrices. We show that principal minors of order at most ℓ, for some invariant ℓ depending only on principal minors of order at most two, uniquely determine principal minors of all orders. In addition, we produce a polynomial-time algorithm that, given access to principal minors, recovers a matrix with those principal minors using only a quadratic number of queries. Furthermore, when principal minors are known only approximately, we present an algorithm that approximately recovers a matrix, and show that the approximation guarantee of this algorithm cannot be improved in general.