Related papers: Matrix Apportionment: General Cases and Apportionm…
Let $K$ be a field, $R=K[x, y]$ the polynomial ring and $\mathcal{M}(K)$ the set of all pairs of square matrices of the same size over $K.$ Pairs $P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $\mathcal{M}(K)$ are called similar if…
The existence and construction of common invariant cones for families of real matrices is considered. The complete results are obtained for 2x2 matrices (with no additional restrictions) and for families of simultaneously diagonalizable…
The problem of matrix factorization motivated by diffraction or elasticity is studied. A powerful tool for analyzing its solutions is introduced, namely analytical continuation formulae are derived. Necessary condition for commutative…
We determine when a matrix is similar to a partial isometry, refining a result of Halmos--McLaughlin.
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…
We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…
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…
We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and…
Matrix approximation is a common tool in machine learning for building accurate prediction models for recommendation systems, text mining, and computer vision. A prevalent assumption in constructing matrix approximations is that the…
The partial transpose map is a linear map widely used quantum information theory. We study the equality condition for a matrix inequality generated by partial transpose, namely $\rank(\sum^K_{j=1} A_j^T \otimes B_j)\le K \cdot…
Given a tract $F$ in the sense of Baker and Bowler and a matrix $A$ with entries in $F$, we define several notions of rank for $A$. In this way, we are able to unify and find conceptually satisfying proofs for various results about ranks of…
Totally equimodular matrices generalize totally unimodular matrices and arise in the context of box-total dual integral polyhedra. This work further explores the parallels between these two classes and introduces foundational building…
Matrix completion is a class of machine learning methods that concerns the prediction of missing entries in a partially observed matrix. This paper studies matrix completion for mixed data, i.e., data involving mixed types of variables…
We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…
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…
Almost-commuting matrices with respect to the normalized Hilbert-Schmidt norm are considered. Normal almost commuting matrices are proved to be near commuting.
Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A and B be upper triangular n-by-n matrices that (i) are not similar to direct sums of matrices of smaller…
Matrix properties are a type of property of categories which includes the ones of being Mal'tsev, arithmetical, majority, unital, strongly unital and subtractive. Recently, an algorithm has been developed to determine implications…
Given a matrix with partitions of its rows and columns and entries from a field, we give the necessary and sufficient conditions that it has a non--singular submatrix with certain number of rows from each row partition and certain number of…
An algebra $\mathcal{A}$ of $n\times n$ complex matrices is said to be \textit{idempotent compressible} if $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. Analogously, $\mathcal{A}$ is said to be…