107 research outputs found
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.Comment: 46 page
Unitary and Euclidean representations of a quiver
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
Canonical matrices for linear matrix problems
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 Belitskii's algorithm for reducing a matrix to a
canonical form, which is the generalization of the Jordan normal form, and
study the set C(m,n) of indecomposable canonical m-by-n matrices. Considering
C(m,n) as a subset in the affine space of m-by-n matrices, we prove that either
C(m,n) consists of a finite number of points and straight lines for every
(m,n), or C(m,n) contains a 2-dimensional plane for a certain (m,n).Comment: 59 page
Estimate of the number of one-parameter families of modules over a tame algebra
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. Respectively, the set of nonisomorphic indecomposable modules
of dimension at most d divides into a finite number f(d,A) of modules and
one-parameter series of modules.
We prove that the number of m-by-n canonical parametric block matrices with a
given partition into blocks is bounded by 4^s, where s is the number of free
entries (which is at most mn), and estimate the number f(d,A).Comment: 23 page
Topological classification of chains of linear mappings
We prove that two chains of linear mappings are topologically isomorphic if
and only if they are linearly isomorphic.Comment: 14 page
Estimate of the number of one-parameter families of modules over a tame algebra
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. Respectively, the set of nonisomorphic indecomposable modules
of dimension at most d divides into a finite number f(d,A) of modules and
one-parameter series of modules.
We prove that the number of m-by-n canonical parametric block matrices with a
given partition into blocks is bounded by 4^s, where s is the number of free
entries (which is at most mn), and estimate the number f(d,A).Comment: 23 page
A canonical form for nonderogatory matrices under unitary similarity
A square matrix is nonderogatory if its Jordan blocks have distinct
eigenvalues. We give canonical forms (i) for nonderogatory complex matrices up
to unitary similarity and (ii) for pairs of complex matrices up to similarity,
in which one matrix has distinct eigenvalues. The types of these canonical
forms are given by undirected and, respectively, directed graphs with no
undirected cycles.Comment: 18 page
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