Vladimir V. Sergeichuk

Canonical matrices for linear matrix problems

Abstract 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 Belitskiis 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×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 .

Canonical forms for complex matrix congruence and ∗congruence

Abstract 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.

Congruences of a square matrix and its transpose

Abstract It is known that any square matrix A over any field is congruent to its transpose: AT=STAS for some nonsingular S; moreover, S can be chosen such that S2=I, that is, S can be chosen to be involutory. We show that A and AT are ∗ congruent over any field F of characteristic not two with involution a↦ a (the involution can be the identity): A T = S T AS for some nonsingular S; moreover, S can be chosen such that S S=I , that is, S can be chosen to be coninvolutory. The short and simple proof is based on Sergeichuks canonical form for ∗ congruence [Math. USSR, Izvestiya 31 (3) (1988) 481]. It follows that any matrix A over F can be represented as A=EB, in which E is coninvolutory and B is symmetric.

Complexity of matrix problems

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

Canonical matrices of bilinear and sesquilinear forms

Abstract Canonical matrices are given for (i) bilinear forms over an algebraically closed or real closed field; (ii) sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and (iii) sesquilinear forms over a field F of characteristic different from 2 with involution (possibly, the identity) up to classification of Hermitian forms over finite extensions of F ; the canonical matrices are based on any given set of canonical matrices for similarity over F . A method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices. This method has its origins in representation theory and was devised in [V.V. Sergeichuk, Classification problems for systems of forms and linear mappings, Math. USSR-Izv. 31 (1988) 481–501].

Tame and wild subspace problems

Assume thatB is a finite-dimensional algebra over an algebraically closed fieldk, Bd=Spec k[(Bd] is the affine algebraic scheme whoseR-points are theB ⊗k k[Bd]-module structures onRd, 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 someFL(d, i) (X); furthermore,FL(d, i) (X) is not isomorphic toFL(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.

Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils

Abstract 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 .

Miniversal deformations of matrices of bilinear forms

Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29–43] constructed miniversal deformations of square complex matrices under similarity; that is, a simp ...

A regularization algorithm for matrices of bilinear and sesquilinear forms

Abstract 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 S ∗ AS = B ⊕ J n 1 ⊕ ⋯ ⊕ J n p , 1 ⩽ n 1 ⩽ ⋯ ⩽ n p , inwhich S is nonsingular and S ∗ = S ∼ T ; B is nonsingular and is determined by A up to ∗congruence; and the ni-by-ni singular Jordan blocks J n i and their multiplicities are uniquely determined by A. We give a regularization algorithm that needs only elementary row operations to construct such a decomposition. If F = C (respectively, F = R ), we exhibit a regularization algorithm that uses only unitary (respectively, real orthogonal) transformations and a reduced form that can be achieved via a unitary *congruence or congruence (respectively, a real orthogonal congruence). The selfadjoint matrix pencil A + λ A ∗ is decomposed by our regularization algorithm into the direct sum S ∗ ( A + λ A ∗ ) S = ( B + λ B ∗ ) ⊕ ( J n 1 + λ J n 1 ∗ ) ⊕ … ⊕ ( J n p + λ J n p ∗ ) with selfadjoint summands.

Unitary and Euclidean representations of a quiver

Abstract 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.