Related papers: On rank-critical matrix spaces

A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A=BB^T. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In…

The problem of finding completely positive matrices with equal cp-rank and rank is considered. We give some easy-to-check sufficient conditions on the entries of a doubly nonnegative matrix for it to be completely positive with equal…

We study the rank weight hierarchy of linear codes which are stable under a linear endomorphism defined over the base field, in particular when the endomorphism is cyclic. In this last case, we give a necessary and sufficient condition for…

This paper studies the inference about linear functionals of high-dimensional low-rank matrices. While most existing inference methods would require consistent estimation of the true rank, our procedure is robust to rank misspecification,…

We show that the space of symmetric matrices of a fixed rank $k$ over a field $K$ of characteristic not equal to $2$ is split Tate. We do this by promoting the point-counting strategy of MacWilliams over finite fields to a filtration of the…

The almost-principal rank characteristic sequence (apr-sequence) of an $n\times n$ symmetric matrix is introduced, which is defined to be the string $a_1 a_2 \cdots a_{n-1}$, where $a_k$ is either $\tt A$, $\tt S$, or $\tt N$, according as…

In [Mulmuley, 1987], Mulmuley gave an algorithm reducing the computation of the matrix rank function to that of determinants, of which the proof for the verification is elementary. In this article, we formalize this argument in the bounded…

A matrix (and any associated linear system) will be referred to as structured if it has a small displacement rank. It is known that the inverse of a structured matrix is structured, which allows fast inversion (or solution), and reduced…

A real $n$-by-$n$ idempotent matrix $A$ with all entries having the same absolute value is called {\it absolutely flat}. We consider the possible ranks of such matrices and herein characterize the triples: size, constant, and rank for which…

Principal matrices of a numerical semigroup of embedding dimension n are special types of $n \times n$ matrices over integers of rank $\leq n - 1$. We show that such matrices and even the pseudo principal matrices of size n must have rank…

Let $R$ and $S$ be two sequences of nonnegative integers in nonincreasing order and with the same sum, and let ${\cal A}(R,S)$ be the class of all $(0,1)$-matrices having row sum $R$ and column sum $S$. For a positive integer $t$, the…

In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of…

We consider the task of updating a matrix function $f(A)$ when the matrix $A\in{\mathbb C}^{n \times n}$ is subject to a low-rank modification. In other words, we aim at approximating $f(A+D)-f(A)$ for a matrix $D$ of rank $k \ll n$. The…

In this paper, we propose and solve a low phase-rank approximation problem, which serves as a counterpart to the well-known low-rank approximation problem and the Schmidt-Mirsky theorem. More specifically, a nonzero complex number can be…

The problem of approximating a matrix by a low-rank one has been extensively studied. This problem assumes, however, that the whole matrix has a low-rank structure. This assumption is often false for real-world matrices. We consider the…

We consider the matrix completion problem with a deterministic pattern of observed entries. In this setting, we aim to answer the question: under what condition there will be (at least locally) unique solution to the matrix completion…

Let X be a nonsingular simply connected projective variety of dimension m, E a rank n vector bundle on X, and L a line bundle on X. Suppose that $S^2(E^{*}) \otimes L$ is an ample vector bundle and that there is a constant even rank $r \ge…

A minor of a matrix is quasi-principal if it is a principal or an almost-principal minor. The quasi principal rank characteristic sequence (qpr-sequence) of an $n\times n$ symmetric matrix is introduced, which is defined as $q_1 q_2 \cdots…

Structured low-rank approximation is the problem of minimizing a weighted Frobenius distance to a given matrix among all matrices of fixed rank in a linear space of matrices. We study exact solutions to this problem by way of computational…

The problem of finding a $k \times k$ submatrix of maximum volume of a matrix $A$ is of interest in a variety of applications. For example, it yields a quasi-best low-rank approximation constructed from the rows and columns of $A$. We show…