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Vladimir V. Sergeichuk

Publications and source records attributed to Vladimir V. Sergeichuk.

At least 55 records · Page 3Linked to original sources
arXiv2010

Block triangular miniversal deformations of matrices and matrix pencils

For each square complex matrix, V. I. Arnold constructed a normal form with the minimal number of parameters to which a family of all matrices B that are close enough to this matrix can be reduced by similarity transformations that smoothly depend on the entries of B. Analogous normal forms were also constructed for families of complex matrix pencils by A. Edelman, E. Elmroth, and B. Kagstrom, and contragredient matrix pencils (i.e., of matrix pairs up to transformations (A,B)-->(S^{-1}AR,R^{-1}BS)) by M. I. Garcia-Planas and V. V. Sergeichuk. In this paper we give other normal forms for families of matrices, matrix pencils, and contragredient matrix pencils; our normal forms are block triangular.

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arXiv2010

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

Docovic and Szechtman, [Proc. Amer. Math. Soc. 133 (2005) 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 the fields of real and complex numbers 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 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.

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arXiv2008

Pairs of mutually annihilating operators

Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations. The classification of (A,B) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636-654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A,B)--> S^{-1}AS, S^{-1}BS).

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arXiv2008

Classification problems for system of forms and linear mappings

We devise a method that reduces the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings. Canonical matrices of (i) bilinear or sesquilinear forms, (ii) pairs of symmetric, skew-symmetric, or Hermitian forms, (iii) isometric or selfadjoint operators on a space with nonsingular symmetric, or skew-symmetric, or Hermitian form are obtained over any field of characteristic not 2 up to classification of Hermitian forms over its finite extensions.

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arXiv2007

Canonical Forms for Unitary Congruence and *Congruence

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, a unitary *congruence canonical form) for complex matrices A such that \bar{A}A (respectively, A^2) is normal. As special cases of our canonical forms, we obtain-in a coherent and systematic way-known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. 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.

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arXiv2007

Generic canonical form of pairs of matrices with zeros

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)--> (SAR,SBL) with nonsingular S, R, L. We prove that almost all of these pairs reduce to the same pair (C, D) from this family, except for pairs whose arbitrary entries are zeros of a certain polynomial. The polynomial and the pair (C D) are constructed by a combinatorial method based on properties of a certain graph.

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arXiv2007

Canonical matrices of isometric operators on indefinite inner product spaces

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: (i) F is an algebraically closed field of characteristic different from 2 or a real closed field, and B is symmetric or skew-symmetric; (ii) F is an algebraically closed field or the skew field of quaternions over a real closed field, and B is Hermitian or skew-Hermitian with respect to any nonidentity involution on F. We use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver. This method was described in [V.V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501].

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arXiv2007

Generic families of matrix pencils and their bifurcation diagrams

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 entries of matrices and got bifurcation diagrams of such families with a small number of parameters. We extend these results to pencils of matrices.

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arXiv2007

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

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 similarity transformations that smoothly depend on the same parameters. We construct analogous normal forms for a family of real matrices and a family of matrix pencils that smoothly depend on parameters, simplifying their normal forms by D. M. Galin [Uspehi Mat. Nauk 27 (1) (1972) 241-242] and by A. Edelman, E. Elmroth, B. Kagstrom [Siam J. Matrix Anal. Appl. 18 (3) (1997) 653-692].

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arXiv2007

Rigid systems of second-order linear differential equations

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 can be reduced by substitutions x(t)=Sy(t), u(t)=Udy(t)/dt+Vy(t)+Pv(t) with nonsingular S and P to each system obtained from it by a small enough perturbation of its matrices A,B,C. We prove that there exists a rigid system if and only if m<n(1+square_root{5})/2, and describe all rigid systems.

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arXiv2007

Congruence of multilinear forms

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 forms.

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arXiv2007

Canonical matrices of bilinear and sesquilinear forms

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 quaternions with any nonidentity involution; and (c) 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. 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, Math. USSR-Izv. 31 (1988) 481-501].

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