94 citations · 912 across the 61 of their papers we have counts for
57 papers · 1 filter
Pairs of commuting nilpotent operators with one-dimensional intersection of kernels and matrices commuting with a Weyr matrix
Vitalij M. Bondarenko, Vyacheslav Futorny, Anatolii P. Petravchuk +1
I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A,B) of commuting nilpotent operators on a vector space contains the problem of classifying an…
Congruence of matrix spaces, matrix tuples, and multilinear maps
Genrich R. Belitskii, Vyacheslav Futorny, Mikhail Muzychuk +1
Two matrix vector spaces are said to be equivalent if for some nonsingular and . These spaces are congruent if . We prove t…
Operators on positive semidefinite inner product spaces
Victor A. Bovdi, Tetiana Klymchuk, Tetiana Rybalkina +2
We give canonical forms of selfadjoint and isometric operators on a complex vector space with scalar product given by a positive semidefinite Hermitian form, and of Hermitian f…
Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form
Jonathan V. Caalim, Vyacheslav Futorny, Vladimir V. Sergeichuk +1
Let be a vector space over a field with scalar product given by a nondegenerate sesquilinear form whose matrix is diagonal in some basis. If ,…
A constructive proof of Pokrzywa's theorem about perturbations of matrix pencils
Vyacheslav Futorny, Tetiana Klymchuk, Vladimir V. Sergeichuk +1
Our purpose is to give new proofs of several known results about perturbations of matrix pencils. Andrzej Pokrzywa (1986) described the closure of orbit of a Kronecker canonical pe…
Lipschitz property for systems of linear mappings and bilinear forms
Abdullah Alazemi, Milica Anđelić, Carlos M. da Fonseca +1
Let G be a graph with undirected and directed edges. Its representation is given by assigning a vector space to each vertex, a bilinear form on the corresponding vector spaces to e…
Wildness for tensors
Vyacheslav Futorny, Joshua A. Grochow, Vladimir V. Sergeichuk
In representation theory, a classification problem is called wild if it contains the problem of classifying matrix pairs up to simultaneous similarity. The latter problem is consid…
Reduction of a pair of skew-symmetric matrices to its canonical form under congruence
V. A. Bovdi, T. G. Gerasimova, M. A. Salim +1
Let be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum \[ (\underline{\underline A},\underline{\und…
Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence
Victor A. Bovdi, Roger A. Horn, Mohamed A. Salim +1
Let be a field of characteristic not , and let be a pair of matrices over , in which is symmetric and is skew-symmetric. A can…
Wildness of the problems of classifying two-dimensional spaces of commuting linear operators and certain Lie algebras
Vyacheslav Futorny, Tetiana Klymchuk, Anatolii P. Petravchuk +1
For each two-dimensional vector space of commuting matrices over a field with at least 3 elements, we denote by the vector space of all $…
Classification of linear mappings between indefinite inner product spaces
Juan Meleiro, Vladimir V. Sergeichuk, Thiago Solovera +1
Let be a linear mapping between vector spaces and over a field or skew field with symmetric, or skew-symmetric, or Hermitian forms $\mathcal…
Generalization of Roth's solvability criteria to systems of matrix equations
Andrii Dmytryshyn, Vyacheslav Futorny, Tetiana Klymchuk +1
W.E. Roth (1952) proved that the matrix equation has a solution if and only if the matrices and $\left[\begin{matrix}A&0…
Specht's criterion for systems of linear mappings
Vyacheslav Futorny, Roger A. Horn, Vladimir V. Sergeichuk
W.Specht (1940) proved that two complex matrices and are unitarily similar if and only if $\operatorname{trace} w(A,A^{\ast}) = \operatorname{trace} w(B,B^{\ast…
Topological classification of systems of bilinear and sesquilinear forms
Carlos M. da Fonseca, Vyacheslav Futorny, Tetiana Rybalkina +1
Let and be two systems consisting of the same vector spaces and bilinear or sesquilinear forms $A_i,B_i:\mathbb C^{n_{k(i)…
Tame systems of linear and semilinear mappings
Tatiana Klimchuk, Dmitry Kovalenko, Tetiana Rybalkina +1
We study systems of linear and semilinear mappings considering them as representations of a directed graph with full and dashed arrows: a representation of is given by assi…
Consimilarity and quaternion matrix equations ,
Tatiana Klimchuk, Vladimir V. Sergeichuk
L.Huang [Linear Algebra Appl. 331 (2001) 21-30] gave a canonical form of a quaternion matrix with respect to consimilarity transformations in which is a…
Regularizing decompositions for matrix pencils and a topological classification of pairs of linear mappings
Vyacheslav Futorny, Tetiana Rybalkina, Vladimir V. Sergeichuk
We give a method for constructing a regularizing decomposition of a matrix pencil, which is formulated in terms of the linear mappings. We prove that two pencils are topologically…
Representations of quivers and mixed graphs
Roger A. Horn, Vladimir V. Sergeichuk
This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras…
Change of the *congruence canonical form of 2-by-2 matrices under perturbations
Vyacheslav Futorny, Lena Klimenko, Vladimir V. Sergeichuk
We study how small perturbations of a 2-by-2 complex matrix can change its canonical form for *congruence. We construct the Hasse diagram for the closure ordering on the set of *co…
Cycles of linear and semilinear mappings
Debora Duarte de Oliveira, Vyacheslav Futorny, Tatiana Klimchuk +2
We give a canonical form of matrices of a cycle of linear or semilinear mapping V_1 --- V_2 --- ... --- V_t --- V_1 in which all V_i are complex vector spaces, each line is an arro…
Systems of subspaces of a unitary space
Vitalij M. Bondarenko, Vyacheslav Futorny, Tatiana Klimchuk +2
For a given poset, we consider its representations by systems of subspaces of a unitary space ordered by inclusion. We classify such systems for all posets for which an explicit cl…
Simultaneous unitary equivalences
Tatiana G. Gerasimova, Roger A. Horn, Vladimir V. Sergeichuk
Let A, B, C, D be given finite sets of pairs of n-by-n complex matrices. We describe an algorithm to determine, with finitely many computations, whether there is a single unitary m…
Remarks on the classification of a pair of commuting semilinear operators
Debora Duarte de Oliveira, Roger A. Horn, Tatiana Klimchuk +1
Gelfand and Ponomarev [Functional Anal. Appl. 3 (1969) 325-326] proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k-tup…
Miniversal deformations of matrices under *congruence and reducing transformations
A. Dmytryshyn, V. Futorny, V. V. Sergeichuk
V.I. Arnold [Russian Math. Surveys 26(2) (1971) 29-43] constructed a miniversal deformation of a square complex matrix under similarity; that is, a simple normal form to which not…