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Clément de Seguins Pazzis

Publications and source records attributed to Clément de Seguins Pazzis.

At least 55 records · Page 3Linked to original sources
arXiv2016

Affine spaces of symmetric or alternating matrices with bounded rank

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. We determine the maximal dimension for an affine subspace of $n$ by $n$ symmetric (or alternating) matrices with entries in $\mathbb{K}$ and with rank less than or equal to $r$. We also classify, up to congruence, the subspaces of maximal dimension among them. This generalizes earlier results of Meshulam, Loewy and Radwan that were previously known only for linear subspaces over fields with large cardinality and characteristic different from $2$.

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arXiv2016

Large spaces of bounded rank matrices revisited

Let $n,p,r$ be positive integers with $n \geq p\geq r$. A rank-$\overline{r}$ subset of $n$ by $p$ matrices (with entries in a field) is a subset in which every matrix has rank less than or equal to $r$. A classical theorem of Flanders states that the dimension of a rank-$\overline{r}$ linear subspace must be less than or equal to $nr$, and it characterizes the spaces with the critical dimension $nr$. Linear subspaces with dimension close to the critical one were later studied by Atkinson, Lloyd and Beasley over fields with large cardinality; their results were recently extended to all fields. Using a new method, we obtain a classification of rank-$\overline{r}$ affine subspaces with large dimension, over all fields. This classification is then used to double the range of (large) dimensions for which the structure of rank $\overline{r}$-linear subspaces is known for all fields.

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arXiv2016

Range-compatible homomorphisms on spaces of symmetric or alternating matrices

Let $U$ and $V$ be finite-dimensional vector spaces over an arbitrary field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space $\mathcal{L}(U,V)$ of all linear maps from $U$ to $V$. A map $F : \mathcal{S} \rightarrow V$ is called range-compatible when it satisfies $F(s) \in \mathrm{im}(s)$ for all $s \in \mathcal{S}$. Among the range-compatible maps are the so-called local ones, that is the maps of the form $s \mapsto s(x)$ for a fixed vector $x$ of $U$. In recent works, we have classified the range-compatible group homomorphisms on $\mathcal{S}$ when the codimension of $\mathcal{S}$ in $\mathcal{L}(U,V)$ is small. In the present article, we study the special case when $\mathcal{S}$ is a linear subspace of the space $S_n(\mathbb{K})$ of all $n$ by $n$ symmetric matrices: we prove that if the codimension of $\mathcal{S}$ in $S_n(\mathbb{K})$ is less than or equal to $n-2$, then every range-compatible homomorphism on $\mathcal{S}$ is local provided that $\mathbb{K}$ does not have characteristic $2$. With the same assumption on the codimension of $\mathcal{S}$, we also classify the range-compatible homomorphisms on $\mathcal{S}$ when $\mathbb{K}$ has characteristic $2$. Finally, we prove that if $\mathcal{S}$ is a linear subspace of the space $A_n(\mathbb{K})$ of all $n$ by $n$ alternating matrices with entries in $\mathbb{K}$, and the codimension of $\mathcal{S}$ is less than or equal to $n-3$, then every range-compatible homomorphism on $\mathcal{S}$ is local.

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arXiv2015

Primitive spaces of matrices with upper rank two over the field with two elements

For fields with more than $2$ elements, the classification of the vector spaces of matrices with rank at most $2$ is already known. In this work, we complete that classification for the field $\mathbb{F}_2$. We apply the results to obtain the classification of triples of locally linearly dependent operators over $\mathbb{F}_2$, the classification of the $3$-dimensional subspaces of $\text{M}_3(\mathbb{F}_2)$ in which no matrix has a non-zero eigenvalue, and the classification of the $3$-dimensional affine spaces that are included in the general linear group $\text{GL}_3(\mathbb{F}_2)$.

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arXiv2015

Hua's fundamental theorem of geometry of rectangular matrices over EAS division rings

The fundamental theorem of geometry of rectangular matrices describes the general form of bijective maps on the space of all $m\times n$ matrices over a division ring $\mathbb{D}$ which preserve adjacency in both directions. This result proved by Hua in the nineteen forties has been recently improved in several directions. One can study such maps without the bijectivity assumption or one can try to get the same conclusion under the weaker assumption that adjacency is preserved in one direction only. And the last possibility is to study maps acting between matrix spaces of different sizes. The optimal result would describe maps preserving adjacency in one direction only acting between spaces of rectangular matrices of different sizes in the absence of any regularity condition (injectivity or surjectivity). A division ring is said to be EAS if it is not isomorphic to any proper subring. It has been known before that it is possible to construct adjacency preserving maps with wild behaviour on matrices over division rings that are not EAS. For matrices over EAS division rings it has been recently proved that adjacency preserving maps acting between matrix spaces of different sizes satisfying a certain weak surjectivity condition are either degenerate or of the expected simple standard form. We will remove this weak surjectivity assumption, thus solving completely the long standing open problem of the optimal version of Hua's theorem.

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arXiv2015

Lines of full rank matrices in large subspaces

Let $n$ and $p$ be non-negative integers with $n \geq p$, and $S$ be a linear subspace of the space of all $n$ by $p$ matrices with entries in a field $\mathbb{K}$. A classical theorem of Flanders states that $S$ contains a matrix with rank $p$ whenever $\mathrm{codim} S <n$. In this article, we prove the following related result: if $\mathrm{codim} S<n-1$, then, for any non-zero $n$ by $p$ matrix $N$ with rank less than $p$, there exists a line that is directed by $N$, has a common point with $S$ and contains only rank $p$ matrices.

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arXiv2015

Quasi-range-compatible affine maps on large operator spaces

Let $U$ and $V$ be finite-dimensional vector spaces over an arbitrary field, and $\mathcal{S}$ be a subset of the space $\mathcal{L}(U,V)$ of all linear maps from $U$ to $V$. A map $F : \mathcal{S} \rightarrow V$ is called range-compatible when it satisfies $F(s) \in \mathrm{im}(s)$ for all $s \in \mathcal{S}$; it is called quasi-range-compatible when the condition is only assumed to apply to the operators whose range does not include a fixed $1$-dimensional linear subspace of $V$. Among the range-compatible maps are the so-called local maps $s \mapsto s(x)$ for fixed $x \in U$. Recently, the range-compatible group homomorphisms on $\mathcal{S}$ were classified when $\mathcal{S}$ is a linear subspace of small codimension in $\mathcal{L}(U,V)$. In this work, we consider several variations of that problem: we investigate range-compatible affine maps on affine subspaces of linear operators; when $\mathcal{S}$ is a linear subspace, we give the optimal bound on its codimension for all quasi-range-compatible homomorphisms on $\mathcal{S}$ to be local. Finally, we give the optimal upper bound on the codimension of an affine subspace $\mathcal{S}$ of $\mathcal{L}(U,V)$ for all quasi-range-compatible affine maps on it to be local.

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arXiv2015

The Flanders theorem over division rings

Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the $\mathbb{F}$-vector space $M_{n,p}(\mathbb{D})$ in which every matrix has rank less than or equal to $r$. Using a new method, we prove that $\dim_{\mathbb{F}} \mathcal{V} \leq \max(n,p)\,rd$ and we characterize the spaces for which equality holds. This extends a famous theorem of Flanders which was known only for fields.

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arXiv2015

Continuous space-time transformations

We prove that every continuous map acting on the four-dimensional Minkowski space and preserving light cones in one direction only is either a Poincaré similarity, that is, a product of a Lorentz transformation and a dilation, or it is of a very special degenerate form. In the presence of the continuity assumption the main tool in the proof is a basic result from the homotopy theory of spheres.

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arXiv2014

LDB division algebras

An LDB division algebra is a triple $(A,\star,\bullet)$ in which $\star$ and $\bullet$ are regular bilinear laws on the finite-dimensional non-zero vector space $A$ such that $x \star (x \bullet y)$ is a scalar multiple of $y$ for all vectors $x$ and $y$ of $A$. This algebraic structure has been recently discovered in the study of the critical case in Meshulam and \v Semrl's estimate of the minimal rank in non-reflexive operator spaces. In this article, we obtain a constructive description of all LDB division algebras over an arbitrary field together with a reduction of the isotopy problem to the similarity problem for specific types of quadratic forms over the given field. In particular, it is shown that the dimension of an LDB division algebra is always a power of $2$, and that it belongs to $\{1,2,4,8\}$ if the characteristic of the underlying field is not $2$.

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arXiv2014

Local linear dependence seen through duality I

A vector space S of linear operators between vector spaces U and V is called locally linearly dependent (in abbreviated form: LLD) when every vector x of U is annihilated by a non-zero operator in S. A duality argument bridges the theory of LLD spaces to the one of vector spaces of non-injective operators. This new insight yields a unified approach to rediscover basic LLD theorems and obtain many additional ones thanks to the power of formal matrix computations. In this article, we focus on the minimal rank for a non-zero operator in an LLD space. Among other things, we reprove the Bresar-Semrl theorem, which states that an n-dimensional LLD operator space always contains a non-zero operator with rank less than n, and we improve the Meshulam-Semrl theorem that examines the case when no non-zero operator has rank less than n-1. We also tackle the minimal rank problem for a non-zero operator in an n-dimensional operator space that is not algebraically reflexive. A theorem of Meshulam and Semrl states that, for all fields with large enough cardinality, a non-reflexive operator space with dimension n must contain a non-zero operator with rank at most 2n-2. We show that there are infinitely many integers n for which this bound is optimal for general infinite fields. Moreover, under mild cardinality assumptions, we obtain a complete classification of the non-reflexive n-dimensional operator spaces in which no non-zero operator has rank less than 2n-2. This classification involves a new algebraic structure called left-division-bilinearizable (in abbreviated form: LDB) division algebras, which generalize a situation that is encountered with quaternions and octonions and whose systematic study occupies a large part of the present article.

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arXiv2014

Range-compatible homomorphisms over fields with two elements

Let $U$ and $V$ be finite-dimensional vector spaces over a (commutative) field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space $\mathcal{L}(U,V)$ of all linear operators from $U$ to $V$. A map $F : \mathcal{S} \rightarrow V$ is called range-compatible when $F(s) \in \text{Im}(s)$ for all $s \in \mathcal{S}$. In a previous work, we have classified all the range-compatible group homomorphisms provided that $\text{codim}(\mathcal{S}) \leq 2\,\text{dim}(V)-3$, except in the special case when $\mathbb{K}$ has only two elements and $\text{codim}(\mathcal{S}) = 2\,\text{dim}(V)-3$. In this article, we give a thorough treatment of that special case. Our results are partly based upon the recent classification of vector spaces of matrices with rank at most $2$ over $\mathbb{F}_2$. As an application, we classify the $2$-dimensional non-reflexive operator spaces over any field, and the affine subspaces of $\text{M}_{n,p}(\mathbb{K})$ with lower-rank $2$ and codimension $3$.

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arXiv2014

On the minimal rank in non-reflexive operator spaces over finite fields

Let $U$ and $V$ be vector spaces over a field $\mathbb{K}$, and $\mathcal{S}$ be an $n$-dimensional linear subspace of $\mathcal{L}(U,V)$. The space $\mathcal{S}$ is called algebraically reflexive whenever it contains every linear map $g : U \rightarrow V$ such that, for all $x \in U$, there exists $f \in \mathcal{S}$ with $g(x)=f(x)$. A theorem of Meshulam and Šemrl states that if $\mathcal{S}$ is not algebraically reflexive then it contains a non-zero operator $f$ of rank at most $2n-2$, provided that $\mathbb{K}$ has more than $n+2$ elements. In this article, we prove that the provision on the cardinality of the underlying field is unnecessary. To do so, we demonstrate that the above result holds for all finite fields.

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arXiv2014

Commutators from a hyperplane of matrices

Denote by $M_n(K)$ the algebra of $n$ by $n$ matrices with entries in the field $K$. A theorem of Albert and Muckenhoupt states that every trace zero matrix of $M_n(K)$ can be expressed as $AB-BA$ for some pair $(A,B)$ of matrices of $M_n(K)$. Assuming that $n>2$ and that $K$ has more than 3 elements, we prove that the matrices $A$ and $B$ can be required to belong to an arbitrary given hyperplane of $M_n(K)$.

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arXiv2013

Range-compatible homomorphisms on matrix spaces

Let K be a (commutative) field, and U and V be finite-dimensional vector spaces over K. Let S be a linear subspace of the space L(U,V) of all linear operators from U to V. A map F from S to V is called range-compatible when F(s) belongs to the range of s for all s in S. Obvious examples of such maps are the evaluation maps s -> s(x), with x in U. In this article, we classify all the range-compatible group homomorphisms on S provided that the codimension of S in L(U,V) is less than or equal to 2 dim(V)-3, unless this codimension equals 2 dim(V)-3 and K has only two elements. Under those assumptions, it is shown that the linear range-compatible maps are the evaluation maps, and the above upper-bound on the codimension of S is optimal for this result to hold. As an application, we obtain new sufficient conditions for the algebraic reflexivity of an operator space and, with the above conditions on the codimension of S, we give an explicit description of the range-restricting and range-preserving homomorphisms on S.

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arXiv2013

Local linear dependence seen through duality II

A vector space S of linear operators between finite-dimensional vector spaces U and V is called locally linearly dependent (in abbreviate form: LLD) when every vector x in U is annihilated by a non-zero operator in S. By a duality argument, one sees that studying LLD operator spaces amounts to studying vector spaces of matrices with rank less than the number of columns, or, alternatively, vector spaces of non-injective operators. In this article, this insight is used to obtain classification results for LLD spaces of small dimension or large essential range (the essential range being the sum of all the ranges of the operators in S). We show that such classification theorems can be obtained by translating into the context of LLD spaces Atkinson's classification of primitive spaces of bounded rank matrices; we also obtain a new classification theorem for such spaces that covers a range of dimensions for the essential range that is roughly twice as large as that in Atkinson's theorem. In particular, we obtain a classification of all 4-dimensional LLD operator spaces for fields with more than 3 elements (beforehand, such a classification was known only for algebraically closed fields and in the context of primitive spaces of matrices of bounded rank). These results are applied to obtain improved upper bounds for the maximal rank in a minimal LLD operator space.

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