Related papers: Positive semidefinite rank and nested spectrahedra

For tensors of fixed order, we establish three types of upper bounds for the geometric rank in terms of the subrank. Firstly, we prove that, under a mild condition on the characteristic of the base field, the geometric rank of a tensor is…

In this paper we study the preservation of strong stability of strongly continuous semigroups on Hilbert spaces. In particular, we study a situation where the generator of the semigroup has a finite number of spectral points on the…

Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of…

Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal A$ of finite rank multiplicative groups infields of characteristic zero. We…

We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…

We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a…

We study the problem of score and rank monotonicity for spectral ranking methods, such as eigenvector centrality and PageRank, in the case of undirected networks. Score monotonicity means that adding an edge increases the score at both ends…

We define a new average - termed the resolvent average - for positive semidefinite matrices. For positive definite matrices, the resolvent average enjoys self-duality and it interpolates between the harmonic and the arithmetic averages,…

This paper is a continuation of my paper "Lattices of flats for symplectic matroids". We explore geometric constructions originating from the lattice of flats of ranked symplectic matroids. We observe that a ranked symplectic matroid always…

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…

We study the algebraic matroid induced by the ideal of (r+1)-minors of a matrix of variables over a field. This is inherently connected to the bounded-rank matrix completion problem, in which the aim is to complete a partially observed rank…

Rank-constrained matrix problems appear frequently across science and engineering. The convergence analysis of iterative algorithms developed for these problems often hinges on local error bounds, which correlate the distance to the…

A symmetric tensor may be regarded as a partially symmetric tensor in several different ways. These produce different notions of rank for the symmetric tensor which are related by chains of inequalities. By exploiting algebraic tools such…

Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=\pi. The following theorem is then proved: If M is a…

A nonnegative tensor has nonnegative rank at most 2 if and only if it is supermodular and has flattening rank at most 2. We prove this result, then explore the semialgebraic geometry of the general Markov model on phylogenetic trees with…

We describe all inequalities among generalized diagonals in positive semi-definite matrices. These turn out to be governed by a simple partial order on the symmetric group. This provides an analogue of results of Drake, Gerrish, and…

We consider the problem of noisy matrix completion, in which the goal is to reconstruct a structured matrix whose entries are partially observed in noise. Standard approaches to this underdetermined inverse problem are based on assuming…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded…

Using the decomposition of semimagic squares into the associated and balanced symmetry types as a motivation, we introduce an equivalent representation in terms of block-structured matrices. This block representation provides a way of…

We are interested in expanding our understanding of symplectic matroids by exploring the properties of a class of symplectic matroids with a "lattice of flats". Taking a well-behaved family of subdivisions of the cross polytope we obtain a…