Related papers: A Short Note on Kronecker Square Roots
Using the block vec matrix, I give a necessary and sufficient condition for factorization of a matrix into the Kronecker product of two other matrices. As a consequence, I obtain an elementary algorithmic procedure to decide whether a…
We solve the problem of characterizing the Kronecker structure of a matrix pencil obtained by a rank-one perturbation of another matrix pencil. The results hold over arbitrary fields.
Translated into the language of representations of quivers, a challenge in matrix pencil theory is to find sufficient and necessary conditions for a Kronecker representation to be a subfactor of another Kronecker representation in terms of…
Many applications give rise to structured matrix polynomials. The problem of constructing structure-preserving strong linearizations of structured matrix polynomials is revisited in this work and in the forthcoming ones…
We introduce a new family of strong linearizations of matrix polynomials---which we call "block Kronecker pencils"---and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any…
The change of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one has been characterized in terms of the homogeneous invariant factors and the chains of column and row minimal indices of the initial and the…
We prove that the border rank of the Kronecker square of the little Coppersmith-Winograd tensor $T_{cw,q}$ is the square of its border rank for $q > 2$ and that the border rank of its Kronecker cube is the cube of its border rank for $q >…
The square root rank of a nonnegative matrix $A$ is the minimum rank of a matrix $B$ such that $A=B \circ B$, where $\circ$ denotes entrywise product. We show that the square root rank of the slack matrix of the correlation polytope is…
In this paper, we study the asymptotic nonnegative rank of matrices, which characterizes the asymptotic growth of the nonnegative rank of fixed nonnegative matrices under the Kronecker product. This quantity is important since it governs…
We determine the border ranks of tensors that could potentially advance the known upper bound for the exponent $\omega$ of matrix multiplication. The Kronecker square of the small $q=2$ Coppersmith-Winograd tensor equals the $3\times 3$…
We investigate the iterated Kronecker product of a square matrix with itself and prove an invariance property for symmetric subspaces. This motivates the definition of an iterated symmetric Kronecker product and the derivation of an…
This is the first in a series of papers on rank decompositions of the matrix multiplication tensor. In this paper we: establish general facts about rank decompositions of tensors, describe potential ways to search for new matrix…
We investigate the construction of circulant matrices derived from primitive roots over finite fields. Our approach reduces exponential sums to Jacobi sums, thereby establishing explicit connections between character theory and matrix…
The starting point for this work is an identity that relates the number of minimal matrices with prescribed 1-marginals and coefficient sequence to a linear combination of Kronecker coefficients. In this paper we provide a bijection that…
We consider the problem of approximately reconstructing a partially-observed, approximately low-rank matrix. This problem has received much attention lately, mostly using the trace-norm as a surrogate to the rank. Here we study low-rank…
A matrix completion problem is to recover the missing entries in a partially observed matrix. Most of the existing matrix completion methods assume a low rank structure of the underlying complete matrix. In this paper, we introduce an…
Under the action of the general linear group with tensor structure, the ranks of matrices $A$ and $B$ forming an $m \times n$ pencil $A + \lambda B$ can change, but in a restricted manner. Specifically, with every pencil one can associate a…
Let $A$ be a square matrix with a given structure (e.g. real matrix, sparsity pattern, Toeplitz structure, etc.) and assume that it is unstable, i.e. at least one of its eigenvalues lies in the complex right half-plane. The problem of…
By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is…
We study the differential and metric structures of the set of real square roots of a non-singular real matrix, under the assumption that the matrix and its square roots are semi-simple, or symmetric, or orthogonal.