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

Change of the *congruence canonical form of 2-by-2 matrices under perturbations

We study how small perturbations of a 2-by-2 complex matrix can change its canonical form for *congruence. We construct the Hasse diagram for the closure ordering on the set of *congruence classes of 2-by-2 matrices.Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:1105.216

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