17 papers · ranked by Valyu relevance
Pascal Koiran
A tuple (Z1, . . . , Zp) of matrices of size r is said to be a commuting extension of a tuple (A1, . . . , Ap) of matrices of size n < r if the Zi pairwise commute and each Ai sits in the upper left corner of a block decomposition of Zi . This notion was discovered and rediscovered in several contexts including…
Emanuel Malvetti
Given a finite set of matrices of size n × n with entries in an algebraically closed field (such as the complex numbers), our goal is to find a basis in which all matrices take on an (upper) triangular shape, or to conclude that no such basis exists. The answer to this question is, for instance, relevant in determining…
Dan Comănescu
diagonalizable matrices Authors: ['Dan Comănescu'] We prove that the following statements are equivalent: a linear matrix equation with parameters forming a commuting set of diagonalizable matrices is consistent, a certain matrix constructed with the Drazin inverse is a solution of this matrix equation, the attached…
Fabienne Chouraqui
Given two linear transformations, with representing matrices A and B with respect to some bases, it is not clear, in general, whether the Tracy-Singh product of the matrices A and B corresponds to a particular operation on the linear transformations. Nevertheless, it is not hard to show that in the particular case that…
Arijit Mukherjee, Gobinda Sau, Arindam Sutradhar
This article studies the equation $[A,B]^k = {\rm Id}_n$ for matrices over $\mathbb{C}$, characterizing the pairs $(k,n)$ for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. The problem is next generalized to matrix rings $M_n(S)$ over arbitrary unital rings $S$, where a…
Haoze He, Daniel Kressner, Bor Plestenjak
It is well known that a family of ${n}\times{n}$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the n joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to…
Shu Li, Jie Wang, Binfeng Wang, Lin Chen
Commutators are essential in quantum information theory, influencing quantum state symmetries and information storage robustness. This paper systematically investigates the characteristics of bipartite and multipartite quantum states invariant under local unitary group actions. The results demonstrate that any quantum…
Caden Young
Let $M_n$ be an $n\times n$ random matrix whose entries are independent Rademacher random variables, and put $N=\binom n2$. We prove [ Pr(M_nM_n^T=M_n^TM_n)=2^{-N+O(n)}. ] This gives the sharp exponential order for the probability that a random sign matrix is normal. The lower bound is supplied by symmetric sign…
Dmitrii Pavlov, Bernd Sturmfels, Simon Telen
Gibbs manifolds are images of affine spaces of symmetric matrices under the exponential map. They arise in applications such as optimization, statistics and quantum physics, where they extend the ubiquitous role of toric geometry. The Gibbs variety is the zero locus of all polynomials that vanish on the Gibbs manifold.…
Markus Pettersen, Nicolai Haug, Joakim Bergli, Thomas M. Surowiec + 1 more
A fundamental challenge in neuroscience and AI is understanding how physical space is mapped into neural representations. While artificial neural networks can generate brain-like spatial representations, such as place and grid cells, their “black-box” nature makes it difficult to determine if these representations…
Rostam M. Razban, Anupam Banerjee, Lilianne R. Mujica-Parodi, Ivet Bahar
Structure determines function. However, this universal theme in biology has been surprisingly difficult to observe in human brain neuroimaging data. Here, we link structure to function by hypothesizing that brain signals propagate as a Markovian process on an underlying structure. We focus on a metric called the…
John Urschel, Cristina H Amon
We prove that every element of the special linear group can be represented as the product of at most six block unitriangular matrices, and that there exist matrices for which six products are necessary, independent of indexing. We present an analogous result for the general linear group. These results serve as general…
Michael Baake, Jeremy Sumner
\usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{d \leqslant 4}$$\end{document} d ⩽ 4 Authors: ['Michael Baake' 'Jeremy Sumner'] The embedding problem of Markov…
Robert Christian Subroto
Circulant Column Parity Mixers (CCPMs) are a particular type of linear maps, used as the mixing layer in permutation-based cryptographic primitives like Keccak-f (SHA3) and Xoodoo. Although being successfully applied, not much is known regarding their algebraic properties. They are limited to invertibility of CCPMs…
Authors not listed
This paper investigates several chemical systems through the lens of hyperstructures and superhyperstructures. We first review Chemical HyperStructures and Chemical SuperHyperStructures defined by redox-driven hyperoperations on species sets with maximal electromotive force selection. Building on the general (m,n)-…
Umashankara Kelathaya, Manjunatha Prasad Karantha
The reverse order law for outer inverses and the Moore-Penrose inverse is discussed in the context of associative rings. A class of pairs of outer inverses that satisfy reverse order law is determined. The notions of left-star and right-star orders have been extended to the case of arbitrary associative rings with…
Lionel Zoubritzky, François-Xavier Coudert
We present here an open-source Julia library for the topological identification of crystalline materials, with algorithmic and computational improvements over the previously available software in the field, resulting in a speed increase of one order of magnitude. This new algorithm and implementation can therefore be…