94 citations · 912 across the 61 of their papers we have counts for
26 papers · 2 filters
Canonical Forms for Unitary Congruence and *Congruence
Roger A. Horn, Vladimir V. Sergeichuk
We use methods of the general theory of congruence and *congruence for complex matrices--regularization and cosquares-to determine a unitary congruence canonical form (respectively…
Miniversal deformations of chains of linear mappings
T. N. Gaiduk, V. V. Sergeichuk, N. A. Zharko
V.I. Arnold [Russian Math. Surveys, 26 (no. 2), 1971, pp. 29-43] gave a miniversal deformation of matrices of linear operators; that is, a simple canonical form, to which not only…
Generic canonical form of pairs of matrices with zeros
Tatyana N. Gaiduk, Vladimir V. Sergeichuk
We consider a family of pairs of m-by-p and m-by-q matrices, in which some entries are required to be zero and the others are arbitrary, with respect to transformations (A,B)--> (S…
Classification of squared normal operators on unitary and Euclidean spaces
Vyacheslav Futorny, Roger A. Horn, Vladimir V. Sergeichuk
We give a canonical form for a complex matrix, whose square is normal, under transformations of unitary similarity as well as a canonical form for a real matrix, whose square is no…
Generic families of matrix pencils and their bifurcation diagrams
M. Isabel Garcia-Planas, Vladimir V. Sergeichuk
V. I. Arnold [Russian Math. Surveys 26, no. 2, 1971, 29-43] constructed smooth generic families of matrices with respect to similarity transformations depending smoothly on the ent…
Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils
M. Isabel Garcia-Planas, Vladimir V. Sergeichuk
V. I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a simple normal form for a family of complex n-by-n matrices that smoothly depend on parameters with respect to…
Canonical matrices of isometric operators on indefinite inner product spaces
Vladimir V. Sergeichuk
We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases:…
Classification of sesquilinear forms with the first argument on a subspace or a factor space
Vyacheslav Futorny, Vladimir V. Sergeichuk
We give canonical matrices of bilinear or sesquilinear forms UxV-->C, (V/U)xV-->C, in which V is a vector space over the field C of complex numbers and U is its subspace.
Rigid systems of second-order linear differential equations
M. Isabel Garcia-Planas, M. Dolors Magret, Vladimir V. Sergeichuk +1
We say that a system of differential equations d^2x(t)/dt^2=Adx(t)/dt+Bx(t)+Cu(t), in which A and B are m-by-m complex matrices and C is an m-by-n complex matrix, is rigid if it ca…
A regularization algorithm for matrices of bilinear and sesquilinear forms
Roger A. Horn, Vladimir V. Sergeichuk
We give an algorithm that uses only unitary transformations and for each square complex matrix constructs a *congruent matrix that is a direct sum of a nonsingular matrix and singu…
Congruence of multilinear forms
Genrich R. Belitskii, Vladimir V. Sergeichuk
It is known that if A and B are two n-by-n complex matrices and (A,A^T) is simultaneously equivalent to (B,B^T), then A is congruent to B. We extend this statement to multilinear f…
Canonical form of m-by-2-by-2 matrices over a field of characteristic other than two
Genrich Belitskii, Maxim Bershadsky, Vladimir V. Sergeichuk
We give a canonical form of m-by-2-by-2 matrices for equivalence over any field of characteristic not two.
Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms
Vyacheslav Futorny, Roger A. Horn, Vladimir V. Sergeichuk
Tridiagonal canonical forms of square matrices under congruence or *congruence, pairs of symmetric or skew-symmetric matrices under congruence, and pairs of Hermitian matrices unde…
Canonical matrices of bilinear and sesquilinear forms
Roger A. Horn, Vladimir V. Sergeichuk
Canonical matrices are given for (a) bilinear forms over an algebraically closed or real closed field; (b) sesquilinear forms over an algebraically closed field and over real quate…
Congruences of a square matrix and its transpose
Roger A. Horn, Vladimir V. Sergeichuk
It is known that any square matrix over any field F is congruent to its transpose. We show that they are also *congruent with respect to any nonidentity involution on F.
Complexity of matrix problems
Genrich R. Belitskii, Vladimir V. Sergeichuk
In representation theory, the problem of classifying pairs of matrices up to simultaneous similarity is used as a measure of complexity; classification problems containing it are c…
Canonical matrices for linear matrix problems
Vladimir V. Sergeichuk
We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct…
Normal form of m-by-n-by-2 matrices for equivalence
Genrich Belitskii, Maxim Bershadsky, Vladimir V. Sergeichuk
We give a canonical form of m-by-2-by-2 spatial matrices for equivalence over any field.
Canonical forms for complex matrix congruence and *congruence
Roger A. Horn, Vladimir V. Sergeichuk
Canonical forms for congruence and *congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347-353], based on Sergeichuk's pape…
Computation of the canonical form for the matrices of chains and cycles of linear mappings
Vladimir V. Sergeichuk
Paul Van Dooren [Linear Algebra Appl. 27 (1979) 103-140] constructed an algorithm for the computation of all irregular summands in Kronecker's canonical form of a matrix pencil. Th…
Estimate of the number of one-parameter families of modules over a tame algebra
Thomas Brüstle, Vladimir V. Sergeichuk
The problem of classifying modules over a tame algebra A reduces to a block matrix problem of tame type whose indecomposable canonical matrices are zero- or one-parameter. Respecti…
Littlewood's algorithm and quaternion matrices
Dennis I. Merino, Vladimir V. Sergeichuk
A strengthened form of Schur's triangularization theorem is given for quaternion matrices with real spectrum (for complex matrices it was given by Littlewood). Littlewood's algorit…
Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild
Genrich Belitskii, Ruvim Lipyanski, Vladimir V. Sergeichuk
We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to congruence, local commutative associative algebras with zero cube radical and square…
The problems of classifying pairs of forms and local algebras with zero cube radical are wild
Genrich Belitskii, Vitalij M. Bondarenko, Ruvim Lipyanski +2
We prove that over an algebraically closed field of characteristic not two the problems of classifying pairs of sesquilinear forms in which the second is Hermitian, pairs of biline…