Vladimir V Sergeichuk | Institute of mathematics NASU, Kyiv (original) (raw)

Papers by Vladimir V Sergeichuk

Research paper thumbnail of Canonical matrices of isometric operators on indefinite inner product spaces

Linear Algebra and Its Applications, 2008

We give canonical matrices of a pair (A, B) consisting of a nondegenerate form B and a linear ope... more 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:

Research paper thumbnail of Generic families of matrix pencils and their bifurcation diagrams

Linear Algebra and Its Applications, 2001

V. I. Arnold ("On matrices depending on parameters", Russian Math. Surveys 26, no. 2, 1971, 29-43... more V. I. Arnold ("On matrices depending on parameters", Russian Math. Surveys 26, no. 2, 1971, 29-43) constructed smooth generic families of matrices with respect to similarity transformations depending smoothly on the entries of matrices and got bifurcation diagrams of such families with a small number of parameters. We extend these results to pencils of matrices.

Research paper thumbnail of Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils

Linear Algebra and Its Applications, 1999

For a family of linear operators A( λ) : U → U over C that smoothly depend on parameters λ = (λ 1... more For a family of linear operators A( λ) : U → U over C that smoothly depend on parameters λ = (λ 1 , . . . , λ k ), V. I. Arnold obtained the simplest normal form of their matrices relative to a smoothly depending on λ change of a basis in U . We solve the same problem for a family of linear operators A( λ) : U → U over R, for a family of pairs of linear mappings A( λ) : U → V, B( λ) : U → V over C and R, and for a family of pairs of counter linear mappings A( λ) : U → V, B( λ) : V → U over C and R. This is the authors' version of a work that was published in Linear Algebra Appl. 302-303 (1999) 45-61.

Research paper thumbnail of Tame and wild subspace problems

Ukrainian Mathematical Journal, 1993

Assume thatB is a finite-dimensional algebra over an algebraically closed fieldk, B d =Spec k[(B ... more Assume thatB is a finite-dimensional algebra over an algebraically closed fieldk, B d =Spec k[(B d ] is the affine algebraic scheme whoseR-points are theB ⊗k k[Bd]-module structures onR d, and Md is a canonical B⊗k k[Bd]-module supported by k[Bd]d. Further, say that an affine subscheme Ν of Bd isclass true if the functor Fgn ∶ X → Md ⊗k[B] X induces an injection between the sets of isomorphism classes of indecomposable finite-dimensional modules over k[Ν] andB. If Bd contains a class-true plane for somed, then the schemes Be contain class-true subschemes of arbitrary dimensions. Otherwise, each Bd contains a finite number of classtrue puncture straight linesL(d, i) such that for eachn, almost each indecomposableB-module of dimensionn is isomorphic to someF L(d, i) (X); furthermore,F L(d, i) (X) is not isomorphic toF L(l, j) (Y) if(d, i) ≠ (l, j) andX ≠ 0. The proof uses a reduction to subspace problems, for which an inductive algorithm permits us to prove corresponding statements.

Research paper thumbnail of Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild

Linear Algebra and Its Applications, 2005

Let F be a field of characteristic different from 2. It is shown that the problems of classifying... more Let F be a field of characteristic different from 2. It is shown that the problems of classifying (i) local commutative associative algebras over F with zero cube radical, (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and (iii) finite p-groups of exponent p with central commutator subgroup of order p 3 are hopeless since each of them contains

Research paper thumbnail of Canonical matrices for linear matrix problems

We consider a large class of matrix problems, which includes the problem of classifying arbitrary... more 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 Belitskiȋ's algorithm for reducing a matrix to a canonical form, which is the generalization of the Jordan normal form, and study the set C mn of indecomposable canonical m × n matrices. Considering C mn as a subset in the affine space of m-by-n matrices, we prove that either C mn consists of a finite number of points and straight lines for every m × n, or C mn contains a 2-dimensional plane for a certain m × n.

Research paper thumbnail of Complexity of matrix problems

Linear Algebra and Its Applications, 2003

In representation theory, the problem of classifying pairs of matrices up to simultaneous similar... more 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 called wild problems. We show in an explicit form that this problem contains all classification matrix problems given by quivers or posets. Then we prove that it does not contain (but is contained in) the problem of classifying threevalent tensors. Hence, all wild classification problems given by quivers or posets have the same complexity; moreover, a solution of any one of these problems implies a solution of each of the others. The problem of classifying three-valent tensors is more complicated.

Research paper thumbnail of The problems of classifying pairs of forms and local algebras with zero cube radical are wild

Linear Algebra and Its Applications, 2005

This is the authors' version of a work that was published in Linear Algebra Appl. 402 (2005) 135-... more This is the authors' version of a work that was published in Linear Algebra Appl. 402 (2005) 135-142. 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 bilinear forms in which the second is symmetric (skew-symmetric), and local algebras with zero cube radical and square radical of dimension 2 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.

Research paper thumbnail of Canonical form of m-by-2-by-2 matrices over a field of characteristic other than two

Linear Algebra and Its Applications, 2006

We give a canonical form of m × 2 × 2 matrices for equivalence over any field of characteristic n... more We give a canonical form of m × 2 × 2 matrices for equivalence over any field of characteristic not two.

Research paper thumbnail of Canonical matrices of bilinear and sesquilinear forms

Linear Algebra and Its Applications, 2008

• bilinear forms over an algebraically closed or real closed field;

Research paper thumbnail of Canonical forms for complex matrix congruence and ∗congruence

Linear Algebra and Its Applications, 2006

Canonical forms for congruence and *congruence of square complex matrices were given by Horn and ... more 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 paper [Math. USSR, Izvestiya 31 (3) (1988) 481-501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms. We use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew-symmetric matrices as well as canonical forms for simultaneous *congruence of pairs of complex Hermitian matrices.

Research paper thumbnail of Unitary and Euclidean representations of a quiver

Linear Algebra and Its Applications, 1998

A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (... more A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations of canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension.

Research paper thumbnail of Matrices that are self-congruent only via matrices of determinant one

Linear Algebra and Its Applications, 2009

2853-2863] considered a vector space V endowed with a bilinear form. They proved that all isometr... more 2853-2863] considered a vector space V endowed with a bilinear form. They proved that all isometries of V over a field F of characteristic not 2 have determinant 1 if and only if V has no orthogonal summands of odd dimension (the case of characteristic 2 was also considered). Their proof is based on Riehm's classification of bilinear forms. Coakley, Dopico, and Johnson [Linear Algebra Appl. 428 (2008) 796-813] gave another proof of this criterion over R and C using Thompson's canonical pairs of symmetric and skew-symmetric matrices for congruence. Let M be the matrix of the bilinear form on V . We give another proof of this criterion over F using our canonical matrices * This is the authors version of a work that was published in Linear Algebra Appl. 431 (2009Appl. 431 ( ) 1620Appl. 431 ( -1632 for congruence and obtain necessary and sufficient conditions involving canonical forms of M for congruence, of (M T , M ) for equivalence, and of M −T M (if M is nonsingular) for similarity.

Research paper thumbnail of Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms

Journal of Algebra, 2008

Tridiagonal canonical forms of square matrices under congruence or *congruence, pairs of symmetri... more Tridiagonal canonical forms of square matrices under congruence or *congruence, pairs of symmetric or skew-symmetric matrices under congruence, and pairs of Hermitian matrices under *congruence are given over an algebraically closed field of characteristic different from 2.

Research paper thumbnail of Unitary and Euclidean representations of a quiver

A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (... more A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations in canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension.

Research paper thumbnail of Classification of squared normal operators on unitary and Euclidean spaces

We give a canonical form for a complex matrix, whose square is normal, under transformations of u... more 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 normal, under transformations of orthogonal similarity.

Research paper thumbnail of Congruences of a square matrix and its transpose

Linear Algebra and Its Applications, 2004

It is known that any square matrix A over any field is congruent to its transpose: A T = S T AS f... more It is known that any square matrix A over any field is congruent to its transpose: A T = S T AS for some nonsingular S; moreover, S can be chosen such that S 2 = I, that is, S can be chosen to be involutory. We show that A and A T are *congruent over any field F of characteristic not two with involution a →ā (the involution can be the identity): A T =S T AS for some nonsingular S; moreover, S can be chosen such thatSS = I, that is, S can be chosen to be coninvolutory. The short and simple proof is based on Sergeichuk's canonical form for *congruence [Math. USSR, Izvestiya 31 (3) (1988) 481-501]. It follows that any matrix A over F can be represented as A = EB, in which E is coninvolutory and B is symmetric.

Research paper thumbnail of Remarks on the classification of a pair of commuting semilinear operators

Gelfand and Ponomarev [Functional Anal. Appl. 3 (1969) 325-326] proved that the problem of classi... more 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-tuples of linear operators for any k. We prove an analogous statement for semilinear operators.

Research paper thumbnail of A regularization algorithm for matrices of bilinear and sesquilinear forms

Linear Algebra and Its Applications, 2006

Over a field or skew field F with an involution a → a (possibly the identity involution), each si... more Over a field or skew field F with an involution a → a (possibly the identity involution), each singular square matrix A is *congruent to a direct sum

Research paper thumbnail of Canonical forms for unitary congruence and *congruence

Linear & Multilinear Algebra, 2009

We show that the classification problems under (a) unitary *congruence when A^3 is normal, and (b... more We show that the classification problems under (a) unitary *congruence when A^3 is normal, and (b) unitary congruence when A\bar{A}A is normal, are both unitarily wild, so there is no reasonable hope that a simple solution to them can be found.

Research paper thumbnail of Canonical matrices of isometric operators on indefinite inner product spaces

Linear Algebra and Its Applications, 2008

We give canonical matrices of a pair (A, B) consisting of a nondegenerate form B and a linear ope... more 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:

Research paper thumbnail of Generic families of matrix pencils and their bifurcation diagrams

Linear Algebra and Its Applications, 2001

V. I. Arnold ("On matrices depending on parameters", Russian Math. Surveys 26, no. 2, 1971, 29-43... more V. I. Arnold ("On matrices depending on parameters", Russian Math. Surveys 26, no. 2, 1971, 29-43) constructed smooth generic families of matrices with respect to similarity transformations depending smoothly on the entries of matrices and got bifurcation diagrams of such families with a small number of parameters. We extend these results to pencils of matrices.

Research paper thumbnail of Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils

Linear Algebra and Its Applications, 1999

For a family of linear operators A( λ) : U → U over C that smoothly depend on parameters λ = (λ 1... more For a family of linear operators A( λ) : U → U over C that smoothly depend on parameters λ = (λ 1 , . . . , λ k ), V. I. Arnold obtained the simplest normal form of their matrices relative to a smoothly depending on λ change of a basis in U . We solve the same problem for a family of linear operators A( λ) : U → U over R, for a family of pairs of linear mappings A( λ) : U → V, B( λ) : U → V over C and R, and for a family of pairs of counter linear mappings A( λ) : U → V, B( λ) : V → U over C and R. This is the authors' version of a work that was published in Linear Algebra Appl. 302-303 (1999) 45-61.

Research paper thumbnail of Tame and wild subspace problems

Ukrainian Mathematical Journal, 1993

Assume thatB is a finite-dimensional algebra over an algebraically closed fieldk, B d =Spec k[(B ... more Assume thatB is a finite-dimensional algebra over an algebraically closed fieldk, B d =Spec k[(B d ] is the affine algebraic scheme whoseR-points are theB ⊗k k[Bd]-module structures onR d, and Md is a canonical B⊗k k[Bd]-module supported by k[Bd]d. Further, say that an affine subscheme Ν of Bd isclass true if the functor Fgn ∶ X → Md ⊗k[B] X induces an injection between the sets of isomorphism classes of indecomposable finite-dimensional modules over k[Ν] andB. If Bd contains a class-true plane for somed, then the schemes Be contain class-true subschemes of arbitrary dimensions. Otherwise, each Bd contains a finite number of classtrue puncture straight linesL(d, i) such that for eachn, almost each indecomposableB-module of dimensionn is isomorphic to someF L(d, i) (X); furthermore,F L(d, i) (X) is not isomorphic toF L(l, j) (Y) if(d, i) ≠ (l, j) andX ≠ 0. The proof uses a reduction to subspace problems, for which an inductive algorithm permits us to prove corresponding statements.

Research paper thumbnail of Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild

Linear Algebra and Its Applications, 2005

Let F be a field of characteristic different from 2. It is shown that the problems of classifying... more Let F be a field of characteristic different from 2. It is shown that the problems of classifying (i) local commutative associative algebras over F with zero cube radical, (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and (iii) finite p-groups of exponent p with central commutator subgroup of order p 3 are hopeless since each of them contains

Research paper thumbnail of Canonical matrices for linear matrix problems

We consider a large class of matrix problems, which includes the problem of classifying arbitrary... more 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 Belitskiȋ's algorithm for reducing a matrix to a canonical form, which is the generalization of the Jordan normal form, and study the set C mn of indecomposable canonical m × n matrices. Considering C mn as a subset in the affine space of m-by-n matrices, we prove that either C mn consists of a finite number of points and straight lines for every m × n, or C mn contains a 2-dimensional plane for a certain m × n.

Research paper thumbnail of Complexity of matrix problems

Linear Algebra and Its Applications, 2003

In representation theory, the problem of classifying pairs of matrices up to simultaneous similar... more 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 called wild problems. We show in an explicit form that this problem contains all classification matrix problems given by quivers or posets. Then we prove that it does not contain (but is contained in) the problem of classifying threevalent tensors. Hence, all wild classification problems given by quivers or posets have the same complexity; moreover, a solution of any one of these problems implies a solution of each of the others. The problem of classifying three-valent tensors is more complicated.

Research paper thumbnail of The problems of classifying pairs of forms and local algebras with zero cube radical are wild

Linear Algebra and Its Applications, 2005

This is the authors' version of a work that was published in Linear Algebra Appl. 402 (2005) 135-... more This is the authors' version of a work that was published in Linear Algebra Appl. 402 (2005) 135-142. 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 bilinear forms in which the second is symmetric (skew-symmetric), and local algebras with zero cube radical and square radical of dimension 2 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.

Research paper thumbnail of Canonical form of m-by-2-by-2 matrices over a field of characteristic other than two

Linear Algebra and Its Applications, 2006

We give a canonical form of m × 2 × 2 matrices for equivalence over any field of characteristic n... more We give a canonical form of m × 2 × 2 matrices for equivalence over any field of characteristic not two.

Research paper thumbnail of Canonical matrices of bilinear and sesquilinear forms

Linear Algebra and Its Applications, 2008

• bilinear forms over an algebraically closed or real closed field;

Research paper thumbnail of Canonical forms for complex matrix congruence and ∗congruence

Linear Algebra and Its Applications, 2006

Canonical forms for congruence and *congruence of square complex matrices were given by Horn and ... more 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 paper [Math. USSR, Izvestiya 31 (3) (1988) 481-501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms. We use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew-symmetric matrices as well as canonical forms for simultaneous *congruence of pairs of complex Hermitian matrices.

Research paper thumbnail of Unitary and Euclidean representations of a quiver

Linear Algebra and Its Applications, 1998

A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (... more A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations of canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension.

Research paper thumbnail of Matrices that are self-congruent only via matrices of determinant one

Linear Algebra and Its Applications, 2009

2853-2863] considered a vector space V endowed with a bilinear form. They proved that all isometr... more 2853-2863] considered a vector space V endowed with a bilinear form. They proved that all isometries of V over a field F of characteristic not 2 have determinant 1 if and only if V has no orthogonal summands of odd dimension (the case of characteristic 2 was also considered). Their proof is based on Riehm's classification of bilinear forms. Coakley, Dopico, and Johnson [Linear Algebra Appl. 428 (2008) 796-813] gave another proof of this criterion over R and C using Thompson's canonical pairs of symmetric and skew-symmetric matrices for congruence. Let M be the matrix of the bilinear form on V . We give another proof of this criterion over F using our canonical matrices * This is the authors version of a work that was published in Linear Algebra Appl. 431 (2009Appl. 431 ( ) 1620Appl. 431 ( -1632 for congruence and obtain necessary and sufficient conditions involving canonical forms of M for congruence, of (M T , M ) for equivalence, and of M −T M (if M is nonsingular) for similarity.

Research paper thumbnail of Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms

Journal of Algebra, 2008

Tridiagonal canonical forms of square matrices under congruence or *congruence, pairs of symmetri... more Tridiagonal canonical forms of square matrices under congruence or *congruence, pairs of symmetric or skew-symmetric matrices under congruence, and pairs of Hermitian matrices under *congruence are given over an algebraically closed field of characteristic different from 2.

Research paper thumbnail of Unitary and Euclidean representations of a quiver

A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (... more A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations in canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension.

Research paper thumbnail of Classification of squared normal operators on unitary and Euclidean spaces

We give a canonical form for a complex matrix, whose square is normal, under transformations of u... more 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 normal, under transformations of orthogonal similarity.

Research paper thumbnail of Congruences of a square matrix and its transpose

Linear Algebra and Its Applications, 2004

It is known that any square matrix A over any field is congruent to its transpose: A T = S T AS f... more It is known that any square matrix A over any field is congruent to its transpose: A T = S T AS for some nonsingular S; moreover, S can be chosen such that S 2 = I, that is, S can be chosen to be involutory. We show that A and A T are *congruent over any field F of characteristic not two with involution a →ā (the involution can be the identity): A T =S T AS for some nonsingular S; moreover, S can be chosen such thatSS = I, that is, S can be chosen to be coninvolutory. The short and simple proof is based on Sergeichuk's canonical form for *congruence [Math. USSR, Izvestiya 31 (3) (1988) 481-501]. It follows that any matrix A over F can be represented as A = EB, in which E is coninvolutory and B is symmetric.

Research paper thumbnail of Remarks on the classification of a pair of commuting semilinear operators

Gelfand and Ponomarev [Functional Anal. Appl. 3 (1969) 325-326] proved that the problem of classi... more 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-tuples of linear operators for any k. We prove an analogous statement for semilinear operators.

Research paper thumbnail of A regularization algorithm for matrices of bilinear and sesquilinear forms

Linear Algebra and Its Applications, 2006

Over a field or skew field F with an involution a → a (possibly the identity involution), each si... more Over a field or skew field F with an involution a → a (possibly the identity involution), each singular square matrix A is *congruent to a direct sum

Research paper thumbnail of Canonical forms for unitary congruence and *congruence

Linear & Multilinear Algebra, 2009

We show that the classification problems under (a) unitary *congruence when A^3 is normal, and (b... more We show that the classification problems under (a) unitary *congruence when A^3 is normal, and (b) unitary congruence when A\bar{A}A is normal, are both unitarily wild, so there is no reasonable hope that a simple solution to them can be found.