Ranks of matrices with few distinct entries

“…The problem of determining the rank of specific matrices has been of immense interest in extremal combinatorics with applications in theoretical computer science as well-see [1,3,4,6,7,9,10,12]. The question in [2] is motivated by a problem in extremal combinatorics concerning what are called self-bisecting families: a family of subsets F of [n] is called a self-bisecting family if, for any distinct A, B ∈ F, either |A∩B| |A| = 1 2 or |A∩B| |B| = 1 2 , and one seeks to find the maximum size of a self-bisecting family of [n].…”

Section: Introductionmentioning

confidence: 99%

An ensemble of high rank matrices arising from tournaments

Balachandran¹,

Bhattacharya²,

Sankarnarayanan³

2021

Preprint

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Suppose F is a field and let a := (a 1 , a 2 , . . . ) be a sequence of non-zero elements in F. For a n := (a 1 , . . . , a n ), we consider the family M n (a) of n × n symmetric matrices M over F with all diagonal entries zero and the (i, j)th element of M either a i or a j for i < j. In this short paper, we show that all matrices in a certain subclass of M n (a)-which can be naturally associated with transitive tournaments-have rank at least ⌊2n/3⌋ − 1. We also show that if char(F) = 2 and M is a matrix chosen uniformly at random from M n (a), then with high probability rank(M ) ≥ 1 2 − o(1) n.

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“…Let N (d, L) denote the maximum size of a square matrix of rank at most d, whose off-diagonal entries belong to a set L. Many important applications of linear algebra to combinatorics reduce to bounding the size of a matrix with few distinct entries and a given rank, i.e., N (d, L). Recently Bukh [4] obtained some general asymptotic results regarding this problem, in particular, he showed that…”

Section: Introductionmentioning

confidence: 99%

“…It is observed in [4] that using the application-specific structure of a matrix may improve upon the upper bound (1.2). For example, if λ min = 0 is the least eigenvalue with multiplicity m of the (0, 1)-adjacency matrix A, then −1 λ min A + I is positive semidefinite of rank d = n − m. Thus it can be seen as the Gram matrix of a set of n unit vectors in R d with two distinct inner products, i.e., a spherical 2-distance set.…”

Section: Introductionmentioning

confidence: 99%