Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear varieties and matroids with applications to the Cullis' determinant

Alexander Guterman, Andrey Yurkov

TL;DR

This work extends Dieudonné's dimension bound for spaces of matrices that annihilate the ordinary determinant to the rectangular setting via the Cullis determinant $\det_{n k}$. It proves that when $n \ge k+2$ and $k$ is odd, the maximal dimension of such a space is $\dim(V)=(n-1)k$, with a sharp description: $V$ consists precisely of matrices whose alternating row sum is zero. The approach combines linear-variety theory and matroid theory, introducing a correspondence between linear varieties and associated matroids, and showing how projections, restrictions, and contractions interact with the matroids. The results yield a clear classification in the odd-$k$ case and provide an upper bound in general, while pointing to open questions for even $k$ and potential applications to linear preservers of Cullis' determinant.

Abstract

Let $V$ be a vector space of rectangular $n\times k$ matrices annihilating the Cullis' determinant. We show that $\dim(V) \le (n-1)k$, extending Dieudonn{é}'s result on the dimension of vector spaces of square matrices annihilating the ordinary determinant. Furthermore, for certain values of $n$ and $k$, we explicitly describe such vector spaces of maximal dimension. Namely, we establish that if $k$ is odd, $n \ge k + 2$ and $\dim(V) = (n-1)k$, then $V$ is equal to the space of all $n\times k$ matrices $X$ such that alternating row sum of $X$ is equal to zero. Our proofs rely on the following observations from the matroid theory that have an independent interest. First, we provide a notion of matroid corresponding to a given linear variety. Second, we prove that if the linear variety is transformed by projections and restrictions, then the behaviour of the corresponding matroid is expressed in the terms of matroid contraction and restriction. Third, we establish that if $M$ is a matroid, $I^*$ its coindependent set $M|S$ and its restriction on a set $S$, then the union of $I^*\setminus S$ with every cobase of $M|S$ is coindependent set of $M$.

Linear varieties and matroids with applications to the Cullis' determinant

TL;DR

This work extends Dieudonné's dimension bound for spaces of matrices that annihilate the ordinary determinant to the rectangular setting via the Cullis determinant . It proves that when and is odd, the maximal dimension of such a space is , with a sharp description: consists precisely of matrices whose alternating row sum is zero. The approach combines linear-variety theory and matroid theory, introducing a correspondence between linear varieties and associated matroids, and showing how projections, restrictions, and contractions interact with the matroids. The results yield a clear classification in the odd- case and provide an upper bound in general, while pointing to open questions for even and potential applications to linear preservers of Cullis' determinant.

Abstract

Let be a vector space of rectangular matrices annihilating the Cullis' determinant. We show that , extending Dieudonn{é}'s result on the dimension of vector spaces of square matrices annihilating the ordinary determinant. Furthermore, for certain values of and , we explicitly describe such vector spaces of maximal dimension. Namely, we establish that if is odd, and , then is equal to the space of all matrices such that alternating row sum of is equal to zero. Our proofs rely on the following observations from the matroid theory that have an independent interest. First, we provide a notion of matroid corresponding to a given linear variety. Second, we prove that if the linear variety is transformed by projections and restrictions, then the behaviour of the corresponding matroid is expressed in the terms of matroid contraction and restriction. Third, we establish that if is a matroid, its coindependent set and its restriction on a set , then the union of with every cobase of is coindependent set of .
Paper Structure (37 sections, 62 theorems, 160 equations)

This paper contains 37 sections, 62 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Notation and basic definitions
  3. Preliminaries from the theory of the Cullis' determinant
  4. The case if $n + k$ is odd.
  5. Property \ref{['SCSprop2']}.
  6. Property \ref{['SCSprop3']}.
  7. Property \ref{['SCSprop4']}.
  8. The case if $n + k$ is even.
  9. Property \ref{['SCSprop2']}.
  10. Property \ref{['SCSprop3']}.
  11. Property \ref{['SCSprop4']}.
  12. Preliminaries from the theory of linear varieties
  13. \ref{['lem:EqRepEqRank:part2']}
  14. \ref{['lem:EqRepEqRank:part3']}
  15. \ref{['lem:InvertibleLDAS:part2']}
  16. ...and 22 more sections

Key Result

Theorem 1.1

Assume that $n \in {\mathbb N}$ and ${\mathbb F}$ is a field such that $(|{\mathbb F}|, n) \neq (2,2)$. Let $V \subseteq {\mathcal{M}}_n({\mathbb F})$ be a vector space, $A \in {\mathcal{M}}_{n\,k}(\mathbb F)$ and ${\mathsf K} = \{A\} + V$. Then the following statements hold:

Theorems & Definitions (121)

  • Theorem 1.1: Dieudonne1948
  • Theorem 1.2: Cf. Theorem \ref{['thm:MaxDimKEvenAltSumZero']}
  • Theorem 1.3: Cf. Theorem \ref{['thm:ALSDetNKZeroCodimGEK']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.7
  • Definition 2.8: NAKAGAMI2007422, Theorem 13
  • ...and 111 more