Related papers: Small maximal spaces of non-invertible matrices

Let K be an arbitrary (commutative) field with at least three elements. It was recently proven that an affine subspace of M_n(K) consisting only of non-singular matrices must have a dimension lesser than or equal to n(n-1)/2. Here, we…

Given an arbitrary (commutative) field K, let V be a linear subspace of M_n(K) consisting of matrices of rank lesser than or equal to some r<n. A theorem of Atkinson and Lloyd states that, if dim V>nr-r+1 and #K>r, then either all the…

To determine whether an $n\times n$-matrix has rank at most $r$ it suffices to check that the $(r+1)\times (r+1)$-minors have rank at most $r$. In other words, to describe the set of $n\times n$-matrices with the property of having rank at…

The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult;…

For a finite simple graph $G$, say $G$ is of dimension $n$, and write $\dim(G) = n$, if $n$ is the smallest integer such that $G$ can be represented as a unit-distance graph in $\mathbb{R}^n$. Define $G$ to be \emph{dimension-critical} if…

The maximal dimension of commutative subspaces of $M_n(\mathbb{C})$ is known. So is the structure of such a subspace when the maximal dimension is achieved. We consider extensions of these results and ask the following natural questions: If…

We introduce a novel concept of rank for subsets of finite metric spaces E^n_q (the set of all n-dimensional vectors over an alphabet of size q) equipped with the Hamming distance, where the rank R(A) of a subset A is defined as the number…

A linear mapping upon real n-dimensional space, where the dimension n is odd, has a real eigenvalue-eigenvector pair. The corresponding statement for complex vector spaces holds true for any dimension n, but should be easy to demonstrate…

Let V be a linear subspace of M_{n,p}(K) with codimension lesser than n, where K is an arbitrary field and n >=p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K is isomorphic…

A classification of semisimple algebras of vector fields on C^N that have a Cartan subalgebra of dimension N is given. The proof uses basic representation theory and the local canonical form of semisimple Lie algebras of vector fields.

A k-ranking of a graph G is a labeling of the vertices of G with values from {1,...,k} such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for…

The $X$-rank of a point $p$ in projective space is the minimal number of points of an algebraic variety $X$ whose linear span contains $p$. This notion is naturally submultiplicative under tensor product. We study geometric conditions that…

A k-ranking of a graph G is a labeling of the vertices of G with values from {1,...,k} such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for…

We use representation theory to construct spaces of matrices of constant rank. These spaces are parametrized by the natural representation of the general linear group or the symplectic group. We present variants of this idea, with more…

In this note we prove two extensions of a recent combinatorial characterization due to Li, Qiao, Wigderson, Wigderson and Zhang (arXiv:2206.04815) of the maximal dimension of bounded rank subspaces of the graphical matrix space associated…

Let $A$ be a real matrix. The term rank of $A$ is the smallest number $t$ of lines (that is, rows or columns) needed to cover all the nonzero entries of $A$. We prove a conjecture of Li et al. stating that, if the rank of $A$ exceeds $t-3$,…

Let K be an arbitrary (commutative) field and L be an algebraic closure of it. Let V be a linear subspace of M_n(K), with n>2. We show that if every matrix of V has at most one eigenvalue in K, then dim V<=1+n(n-1)/2. If every matrix of V…

Let G be a connected simple algebraic group over an algebraically closed field k of characteristic p > 0, and g := Lie(G). We additionally assume that G is standard and is of type An. Motivated by the investigation of the geometric…

A k-ranking of a graph G is a labeling of the vertices of G with values from 1,...,k such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for…

The focus of the paper is on the maximal dimension of affine subspaces of nilpotent $n \times n $ matrices with fixed rank. In particular we obtain two results in the "border" cases rank equal to $n-1$ and rank equal to $1$.