?
Maximal generalized rank in graphical matrix spaces
Israel Journal of Mathematics. 2023. Vol. 256. No. 1. P. 297 – 309.
Spiridonov I., Meshulam R., Гутерман А.
Sharma H., Shirokov D., Advances in Applied Clifford Algebras 2025 Vol. 35 Article 44
Commutative analogues of Clifford algebras are algebras defined in the same way as Clifford algebras except that their generators commute with each other, in contrast to Clifford algebras in which the generators anticommute. In this paper, we solve the problem of finding multiplicative inverses in commutative analogues of Clifford algebras by introducing a matrix representation ...
Added: October 2, 2025
Shirokov D., Advances in Applied Clifford Algebras 2025 Vol. 35 Article 25
We discuss a generalization of Clifford algebras known as generalized Clifford algebras (in particular, ternary Clifford algebras). In these objects, we have a fixed higher-degree form (in particular, a ternary form) instead of a quadratic form in ordinary Clifford algebras. We present a natural realization of unitary Lie groups, which are important in physics and ...
Added: May 20, 2025
Shirokov D., , in: Advances in Computer Graphics: 41st Computer Graphics International Conference, CGI 2024, Geneva, Switzerland, July 1–5, 2024, Proceedings, Part IIIVol. 15340.: Springer, 2025. P. 336–348.
Added: April 1, 2025
Spiridonov I., Guterman A., Linear Algebra and its Applications 2024 Vol. 680 P. 325–340
Added: October 3, 2024
Costara C., A. E. Guterman, A. M. Maksaev et al., Linear Algebra and its Applications 2023 Vol. 666 P. 129–143
In this paper, we characterize the automorphisms of the total graph for the ring M_n of matrices of order n≥2 over any field with at least 3 elements. To do this, we apply the technique of maps preserving matrix invariants; in particular, as an intermediate step, we characterize pairs of surjective maps φ_1,φ_2:M_n →M_n such that A+B is singular if and only if φ_1(A)+φ_2(B) is ...
Added: April 11, 2023
A.E. Guterman, A.M. Maksaev, V.V. Promyslov, Linear Algebra and its Applications 2022 Vol. 644 P. 1–27
Let F be an algebraically closed field and Mn be the n × n matrix algebra over F. A total graph of the full matrix
algebra is the graph with Mn as vertices, and two distinct matrices A, B are adjacent if and only if A+B is singular. The
characterization of all the automorphisms of the total ...
Added: June 13, 2022
Shirokov D., Computational and Applied Mathematics 2021 Vol. 40 P. 1–29
In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension $n$. The formulas involve only ...
Added: July 15, 2021
Guterman A. E., Spiridonov I.A., Linear Algebra and its Applications 2020 Vol. 599 P. 140–155
Let $M_{n}(\mathbb{F})$ denote the set of square matrices of size $n$ over a field $\mathbb{F}$ with characteristics different from two. We say that the map $f: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ is additive if $f(A+B) = f(A) + f(B)$ for all $A, B \in M_{n}(\mathbb{F})$. The main goal of this paper is to prove that for $n>2$ ...
Added: November 9, 2020
Shitov Y., Linear Algebra and its Applications 2018 Vol. 554 P. 49–50
We prove that the determinant of an n x n 01-matrix with at most n+k non-zero entries does not exceed α^k with α=4^(1/3)≈1.316074. ...
Added: January 30, 2019
Burman Y. M., Journal of Algebraic Combinatorics 2019 Vol. 50 No. 4 P. 427–446
The classical matrix-tree theorem discovered by G.Kirchhoff in 1847
expresses the principal minor of the (n x n) Laplace matrix as a
sum of monomials of matrix elements indexed by directed trees with n
vertices. We prove, for any k >= n, a three-parameter family of
identities between degree k polynomials of matrix elements of the
Laplace matrix. For k=n and special values of ...
Added: October 18, 2018
Shitov Y., Linear Algebra and its Applications 2018 Vol. 554 P. 49–50
We prove that the determinant of an n-by-n 0-1 matrix with at most n + k non-zero entries does not exceed α^k with α ≈ 1.316074. ...
Added: September 26, 2018
M.N.Vyalyi, Babenko A. V., Computational Mathematics and Mathematical Physics 2017 Vol. 57 No. 2 P. 362–371
The problem of linear classification of the parity of permutation matrices is studied. This problem is related to the analysis of complexity of a class of algorithms designed for computing the permanent of a matrix that generalizes the Kasteleyn algorithm. Exponential lower bounds on the magnitude of the coefficients of the functional that classifies the ...
Added: October 16, 2017
Rotmistrov A., Popova P., Социология: методология, методы, математическое моделирование 2016 № 43 С. 63–99
The focus of this article is the methodological aspect of political activism determinants identifying; specifically variants of handling with categorical predictors which hypothetically explain the level of activism. When using regression for explaining the issue, one may transform such predictors into dummy variables. Such a popular solution makes the model bulky and causes troubles with ...
Added: March 15, 2017
Alexander Guterman, Yaroslav Shitov, Linear Algebra and its Applications 2016 Vol. 498 P. 326–348
We consider rank functions important in tropical linear algebra: the tropical rank, equal to the topological dimension of the tropical linear span of the columns of a given matrix, the factor rank, equal to the smallest number of vectors containing these columns in their span, the Kapranov rank, related to problems of tropical algebraic geometry, ...
Added: August 10, 2015
Beasley L., Guterman A., Shitov Y., Journal of Algebra 2015 Vol. 433 P. 168–182
Among different rank functions on tropical matrices, there is one known as tropical rank which is a lower bound for any other. Here we introduce a new concept (for being opposed to tropical rank, it is called arctic) which gives an upper bound for other ranks. Our definition is based on the perimeter notion previously ...
Added: April 14, 2015
Shitov Y., European Journal of Combinatorics 2014 Vol. 42 P. 107–111
Let A be a real matrix. The term rank of A is the smallest number t of lines (that is, rows or columns) needed to cover all the nonzero entries of A. We prove a conjecture of Li et al. stating that, if the rank of A exceeds t-3, there is a rational matrix with ...
Added: June 23, 2014
Yaroslav Shitov, Linear Algebra and its Applications 2013 Vol. 439 No. 8 P. 2500–2502
We present a reduction which shows that the fooling set number, tropical and determinantal ranks of a Boolean matrix are NP-hard to compute. ...
Added: August 11, 2013
Burman Y. M., Ploskonosov A., Trofimova A., / Series math "arxiv.org". 2011. No. 1109.6625.
We calculate determinants of weighted sums of reflections and of (nested) commutators of reflections. The results obtained generalize the matrix-tree theorem by Kirchhoff and the Pfaffian-hypertree theorem by Massbaum and Vaintrob. ...
Added: November 7, 2012