Related papers: A Combinatorial Result on Block Matrices
This paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. This generalizes the standard low-rank matrix completion…
We give sufficient conditions on a matrix A ensuring the existence of a partition of this matrix into two submatrices with extremely small norm of the image of any vector. Under some weak conditions on a matrix A we obtain a partition of A…
In standard clustering problems, data points are represented by vectors, and by stacking them together, one forms a data matrix with row or column cluster structure. In this paper, we consider a class of binary matrices, arising in many…
In this paper, a matrix is said to be prime if the row and column of this matrix are both prime numbers. We establish various necessary and sufficient conditions for developing matrices into the sum of tensor products of prime matrices. For…
An integer matrix $\mathbf{A}$ is $\Delta$-modular if the determinant of each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix of $\mathbf{A}$ has absolute value at most $\Delta$. The study of $\Delta$-modular matrices…
We study the row completion problem of polynomial and rational matrices with partial prescription of the structural data. The prescription of the complete structural data has been solved in Amparan et al., Lin. Alg. Appl. 720 (2025)…
For a given selection of rows and columns from a Fourier matrix, we give a number of tests for whether the resulting submatrix is Hadamard based on the primitive sets of those rows and columns. In particular, we demonstrate that whether a…
Matrices can be augmented by adding additional columns such that a partitioning of the matrix in blocks of rows defines mutually orthogonal subspaces. This augmented system can then be solved efficiently by a sum of projections onto these…
In the paper, the authors present several new relations and applications for the combinatorial sequence that counts the possible partitions of a finite set with the restriction that the size of each block is contained in a given set. One of…
This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then…
We propose necessary and sufficient conditions for an integer matrix to be decomposable in terms of its Hermite normal form. Specifically, to each integer matrix of maximal row rank without columns of zeros, we associate a symmetric whole…
We describe combinatorial properties of the defining row of a circulant Hadamard matrix by exploiting its orthogonality to subsequent rows, and show how to exclude several particular forms of these matrices.
Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank $r$ and with bounded subdeterminants. In particular, we study the…
This paper considers a restriction to non-negative matrix factorization in which at least one matrix factor is stochastic. That is, the elements of the matrix factors are non-negative and the columns of one matrix factor sum to 1. This…
We investigate the existence of heavy columns in binary matrices with distinct rows. A column of an m x n binary matrix is called heavy if the number of ones in it is at least m/2. We introduce two recursive algorithms, A1 and A2, that…
The combinatorial properties of partitions with various restrictions on their hooksets are explored. A connection with numerical semigroups extends current results on simultaneous s/t-cores. Conditions that suffice for a partition to…
The partial transpose of a block matrix M is the matrix obtained by transposing the blocks of M independently. We approach the notion of partial transpose from a combinatorial point of view. In this perspective, we solve some basic…
To a given nonsingular triangular matrix A with entries from a ring, we associate a weighted bipartite graph G(A) and give a combinatorial description of the inverse of A by employing paths in G(A). Under a certain condition, nonsingular…
We call a matrix completely mixable if the entries in its columns can be permuted so that all row sums are equal. If it is not completely mixable, we want to determine the smallest maximal and largest minimal row sum attainable. These…
The main results of this paper are twofold: the first one is a matrix theoretical result. We say that a matriz is superregular if all of its minors that are not trivially zero are nonzero. Given a a times b, a larger than or equal to b,…