arXiv subjects
Publications and source records attributed to Elena Rubei.
The focus of the paper is on the maximal dimension of affine subspaces of nilpotent n x n matrices with fixed rank. In particular we obtain a result in the "border" case rank equal to n-1; we also consider the much simpler case rank equal to 1.
For every $n \in \mathbb{N}$ and every field $K$, let $A(n,K)$ be the vector space of the antisymmetric $(n \times n)$-matrices over $K$. We say that an affine subspace $S$ of $A(n,K)$ has constant rank $r$ if every matrix of $S$ has rank $r$. Define $${\cal A}_{antisym}^K(n;r)= \{ S \;| \; S \; \mbox{\rm affine subspace of $A(n,K)$ of constant rank } r\}$$ $$a_{antisym}^K(n;r) = \max \{\dim S \mid S \in {\cal A}_{antisym}^K(n;r) \}.$$ In this paper we prove the following formulas: for $n \geq 2r +2 $ $$a_{antisym}^{\mathbb{R}}( n; 2r) = (n-r-1) r ;$$ for $n=2r$ $$a_{antisym}^{\mathbb{R}}( n; 2r) =r(r-1) ;$$ for $n=2r+1$ $$a_{antisym}^{\mathbb{R}}( n; 2r) = r(r+1) .$$
Let $α$ be a $p \times q$ interval matrix with $p \geq q$ and with the endpoints of all its entries in the set of the rational numbers. We prove that, if $α$ contains a rank-$r$ real matrix with $r \in \{2, q-2,q-1,q\}$, then it contains a rank-$r$ rational matrix.
For every $m,n \in \mathbb{N}$ and every field $K$, let $M(m \times n, K)$ be the vector space of the $(m \times n)$-matrices over $K$ and let $S(n,K)$ be the vector space of the symmetric $(n \times n)$-matrices over $K$. We say that an affine subspace $S$ of $M(m \times n, K)$ or of $S(n,K)$ has constant rank $r$ if every matrix of $S$ has rank $r$. Define $${\cal A}^K(m \times n; r)= \{ S \;| \; S \; \mbox{\rm affine subsapce of $M(m \times n, K)$ of constant rank } r\}$$ $${\cal A}_{sym}^K(n;r)= \{ S \;| \; S \; \mbox{\rm affine subsapce of $S(n,K)$ of constant rank } r\}$$ $$a^K(m \times n;r) = \max \{\dim S \mid S \in {\cal A}^K(m \times n; r ) \}.$$ $$a_{sym}^K(n;r) = \max \{\dim S \mid S \in {\cal A}_{sym}^K(n,r) \}.$$ In this paper we prove the following two formulas for $r \leq m \leq n$: $$a_{sym}^{\mathbb{R}}(n;r) \leq \left\lfloor \frac{r}{2} \right\rfloor \left(n- \left\lfloor \frac{r}{2} \right\rfloor\right)$$ $$a^{\mathbb{R}}(m \times n;r) = r(n-r)+ \frac{r(r-1)}{2} .$$
A general closed interval matrix is a matrix whose entries are closed connected nonempty subsets of the set of the real numbers, while an interval matrix is defined to be a matrix whose entries are closed bounded nonempty intervals in the set of real numbers. We say that a matrix $A$ with constant entries is contained in a general closed interval matrix $μ$ if and only if, for every $i,j$, we have that $A_{i,j} \in μ_{i,j}$. Rhon characterized full-rank square interval matrices, that is, square interval matrices $μ$ such that every constant matrix contained in $μ$ is nonsingular. In this paper we generalize this result to general closed interval matrices.
The problem of finding the maximal dimension of linear or affine subspaces of matrices whose rank is constant, or bounded below, or bounded above, has attracted many mathematicians from the sixties to the present day. The problem has caught also the attention of algebraic geometers since vector spaces of matrices of constant rank $r$ give rise to vector bundle maps whose images are vector bundles of rank $r$. Moreover there is a link with the so called ``rank metric codes'', since a constant rank $r$ subspace of $K^{n \times n}$ can be viewed as a constant weight $r$ rank metric code; it can be interesting to study also the maximal dimension of the subspaces of $K^{n \times n}$ whose elements have rank in a range $[s,r]$, since such subspaces obviously give rank metric codes with weights in $[s,r]$. In this paper, with the main purpose to get an organic result including the ones on spaces of matrices with constant rank, the ones on spaces of matrices with rank bounded below and the ones on spaces of matrices with rank bounded above and to generalize a previous result on real matrices with constant rank to matrices on a more general field, we study the maximal dimension of affine subspaces of matrices whose rank is between two numbers under mild assumptions on the field. We get also a result on antisymmetric matrices and on matrices in row echelon form.
For every $n \in \mathbb{N}$ and every field $K$, let $N(n,K)$ be the set of the nilpotent $n \times n$ matrices over $K$ and let $D(n,K) $ be the set of the $n \times n$ matrices over $K$ which are diagonalizable over $K$. Moreover, let $R(n) $ be the set of the normal $n \times n$ matrices. In this short note we prove that the maximal dimension of an affine subspace in $N(n,K)$ is $ \frac{n(n-1)}{2}$ and, if the characteristic of the field is zero, an affine not linear subspace in $N(n,K)$ has dimension less than or equal to $ \frac{n(n-1)}{2}-1$. Moreover we prove that the maximal dimension of an affine subspace in $R(n)$ is $n$, the maximal dimension of a linear subspace in $D(n, \mathbb{R})$ is $ \frac{n(n+1)}{2}$, while the maximal dimension of an affine not linear subspace in $D(n, \mathbb{R})$ is $ \frac{n(n+1)}{2} -1$.
Given a positive-weighted simple connected graph with $m$ vertices, labelled by the numbers $1,\ldots,m$, we can construct an $m \times m$ matrix whose entry $(i,j)$, for any $i,j\in\{1,\dots,m\}$, is the minimal weight of a path between $i$ and $j$, where the weight of a path is the sum of the weights of its edges. Such a matrix is called the distance matrix of the weighted graph. There is wide literature about distance matrices of weighted graphs. In this paper we characterize distance matrices of positive-weighted $n$-hypercube graphs. Moreover we show that a connected bipartite $n$-regular graph with order $2^n$ is not necessarily the $n$-hypercube graph. Finally we give a characterization of distance matrices of positive-weighted Petersen graphs.
An interval matrix is a matrix whose entries are intervals in the set of real numbers. We generalize this concept, which has been broadly studied, to other fields. Precisely we define a rational interval matrix to be a matrix whose entries are intervals in the set of the rational numbers. We prove that a (real) interval matrix with endpoints of all its entries in the set of the rational numbers contains a rank-one matrix if and only if contains a rational rank-one matrix and contains a matrix with rank smaller than min{p,q} if and only if it contains a rational matrix with rank smaller than min{p,q}; from these results and from the analogous criterions for (real) inerval matrices, we deduce immediately a criterion to see when a rational interval matrix contains a rank-one matrix and a criterion to see when it is full-rank, that is, all the matrices it contains are full-rank. Moreover, given a field K and a matrix a whose entries are subsets of K, we describe a criterion to find the maximal rank of a matrix contained in a.
An interval matrix is a matrix whose entries are intervals in the set of real numbers. Let $p , q $ be nonzero natural numbers and let $μ=( [m_{i,j}, M_{i,j}])_{i,j}$ be a $p \times q$ interval matrix; given a $p \times q$ matrix $A$ with entries in the set of real numbers, we say that $ A \in μ$ if $a_{i,j} \in [m_{i,j}, M_{i,j}] $ for any $i,j$. We establish a criterion to say if an interval matrix contains a matrix of rank $1$. Moreover we determine the maximum rank of the matrices contained in a given interval matrix. Finally, for any interval matrix $μ$ with no more than $3$ columns, we describe a way to find the range of the ranks of the matrices contained in $μ$.
Let ${\cal G}=(G,w)$ be a weighted graph , that is, a graph $G$ endowed with a function $w$ from the edge set of $G$ to the set of real numbers; for any subset $S$ of the vertex set of $G$, we define $D_S({\cal G})$ to be the minimum of the weights of the subgraphs of $G$ whose vertex set contains $S$; we call $D_S({\cal G})$ a multiweight of ${\cal G}$. Let $X$ be a finite set and let $\{D_S\}_{S \subset X, \; \sharp S \geq 2} $ be a family of positive real numbers. We find necessary and sufficient conditions for the family to be the family of multiweights of a positive-weighted graph with vertex set $X$. Moreover we study the analogous problem for trees. Finally, we find a criterion to say if there exists a nonnegative-weighted tree ${\cal T}$ with leaf set $X$ and such that $D_S ({\cal T})=D_S $ for any $S \subset X$.
Let ${\cal G}=(G,w) $ be a positive-weighted graph, that is a graph $G$ endowed with a function $w$ from the edge set of $G$ to the set of positive real numbers; for any distinct vertices $i,j $, we define $D_{i,j}({\cal G})$ to be the weight of the path in $G$ joining $i$ and $j$ with minimum weight. In this paper we fix a particular class of graphs and we give a criterion to establish whether, given a family of positive real numbers $\{D_I\}_{I \in { \{1,...., n\} \choose 2}}$, there exists a positive-weighted graph ${\cal G} =(G,w) $ in the class we have fixed, with vertex set equal to $\{1,....,n\}$ and such that $D_I ({\cal G}) =D_I$ for any $I \in { \{1,...., n\} \choose 2}$. In particular, the classes of graphs we consider are the following: snakes, caterpillars, polygons, bipartite graphs, complete graphs, planar graphs.
Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$.For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$, let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$ are called $k$-weights of ${\cal T}$. Given a family of real numbers parametrized by the $k$-subsets of $\{1,..., n\}$, $\{D_I\}_{I \in {\{1,...,n\} \choose k}}$, we say that a weighted tree ${\cal T}=(T,w)$ with leaves $1,..., n$ realizes the family if $D_I({\cal T})=D_I$ for any $ I $. In 2006 Levy, Yoshida and Pachter defined, for any positive-weighted tree ${\cal T}=(T,w)$ with $\{1,..., n\}$ as leaf set and any $i, j \in \{1,..., n\}$, the numbers $S_{i,j}$ to be $ \sum_{Y \in {\{1,..., n\} -\{i,j\} \choose k-2}} D_{i,j ,Y}({\cal T}) $; they proved that there exists a positive-weighted tree ${\cal T}' =(T',w')$ such that $D_{i,j}({\cal T}')=S_{i,j}$ for any $i,j \in \{1,..., n\}$ and that this new tree is, in some way, similar to the given one. In this paper, by using the $S_{i,j}$ defined by Levy, Yoshida and Pachter, we characterize families of real numbers parametrized by ${\{1,...,n\} \choose k}$ that are the families of $k$-weights of weighted trees with leaf set equal to $\{1,...., n\}$ and weights of the internal edges positive.
Let $X$ be a finite set. We give criterion to say if a system of trees ${\cal P}=\{T_i\}_i$ with leaf sets $L(T_i) \in {X \choose 5}$ can be amalgamated into a supertree, that is, if there exists a tree $T$ with $L(T)=X$ such that $T$ restricted to $ L(T_i)$ is equal to $T_i$.
Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$. For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$, let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$ are called $k$-weights of ${\cal T}$. Given a family of real numbers parametrized by the $k$-subsets of $\{1,..., n\}$, $\{D_I\}_{I \in {\{1,...,n\} \choose k}}$, we say that a weighted tree ${\cal T}=(T,w)$ with leaves $1,..., n$ realizes the family if $D_I({\cal T})=D_I$ for any $ I $. We give a characterization of the families of real numbers that are realized by some weighted tree.
Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$.For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$,let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$ are called $k$-weights of ${\cal T}$. Given a family of real numbers parametrized by the $k$-subsets of $ \{1,..., n\}$, $\{D_I\}_{I \in {\{1,...,n\} \choose k}}$, we say that a weighted tree ${\cal T}=(T,w)$ with leaves $1,..., n$ realizes the family if $D_I({\cal T})=D_I$ for any $ I $. In [P-S] Pachter and Speyer proved that, if $3 \leq k \leq (n+1)/2$ and $\{D_I\}_{I \in {\{1,...,n\} \choose k}}$ is a family of positive real numbers, then there exists at most one positive-weighted essential tree ${\cal T}$ with leaves $1,...,n$ that realizes the family (where "essential" means that there are no vertices of degree $2$). We say that a tree $P$ is a pseudostar of kind $(n,k)$ if the cardinality of the leaf set is $n$ and any edge of $P$ divides the leaf set into two sets such that at least one of them has cardinality $ \geq k$. Here we show that, if $3 \leq k \leq n-1$ and $\{D_I\}_{I \in {\{1,...,n\} \choose k}}$ is a family of real numbers realized by some weighted tree, then there is exactly one weighted essential pseudostar ${\cal P}=(P,w)$ of kind $(n,k)$ with leaves $1,...,n$ and without internal edges of weight $0$, that realizes the family; moreover we describe how any other weighted tree realizing the family can be obtained from ${\cal P}$. Finally we examine the range of the total weight of the weighted trees realizing a fixed family.
Let ${\cal G}=(G,w)$ be a weighted simple finite connected graph, that is, let $G$ be a simple finite connected graph endowed with a function $w$ from the set of the edges of $G$ to the set of real numbers. For any subgraph $G'$ of $G$, we define $w(G')$ to be the sum of the weights of the edges of $G'$. For any $i,j $ vertices of $G$, we define $D_{\{i,j\}} ({\cal G})$ to be the minimum of the weights of the simple paths of $G$ joining $i$ and $j$. The $D_{\{i,j\}} ({\cal G})$ are called $2$-weights of ${\cal G}$. Let $\{m_I\}_{I \in {\{1,...,n\} \choose 2}}$ and $\{M_I\}_{I \in {\{1,...,n\} \choose 2}}$ be two families of positive real numbers parametrized by the $2$-subsets of $ \{1,..., n\}$ with $m_I \leq M_I$ for any $I$; we study when there exist a positive-weighted graph ${\cal G}$ and an $n$-subset $\{1,..., n\}$ of the set of its vertices such that $D_I ({\cal G}) \in [m_I, M_I] $ for any $I \in {\{1,...,n\} \choose 2}$. Then we study the analogous problem for trees, both in the case of positive weights and in the case of general weights.
A partial matrix is a matrix where only some of the entries are given. We determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given and the minimum rank and the maximum rank of the antisymmetric completions of an antisymmetric partial matrix where only the diagonal blocks are given.