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 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).

59 pages

References in corpus (3)

• Canonical matrices for linear matrix problems
• Classification problems for system of forms and linear mappings
• Unitary and Euclidean representations of a quiver

Cited by in corpus (25)

• Canonical matrices for linear matrix problems
• Complexity of matrix problems
• Computation of the canonical form for the matrices of chains and cycles of linear mappings
• Canonical matrices of isometric operators on indefinite inner product spaces
• Wildness for tensors
• Pairs of mutually annihilating operators
• Representations of quivers and mixed graphs
• Isomorphism problems for tensors, groups, and cubic forms: completeness and reductions
• A canonical form for nonderogatory matrices under unitary similarity
• Remarks on the classification of a pair of commuting semilinear operators
• Neighborhood radius estimation for Arnold's miniversal deformations of complex and $p$-adic matrices
• Matrices that are self-congruent only via matrices of determinant one
• Topological classification of chains of linear mappings
• Estimate of the number of one-parameter families of modules over a tame algebra
• A spectral theory for transverse tensor operators
• Pairs of commuting nilpotent operators with one-dimensional intersection of kernels and matrices commuting with a Weyr matrix
• Operators on positive semidefinite inner product spaces
• Rigid systems of second-order linear differential equations
• Wildness of the problem of classifying nilpotent Lie algebras of vector fields in four variables
• Classification of G_2-orbits for pairs of octonions
• On triangular similarity of nilpotent triangular matrices
• On similar matrices over the dual numbers
• A constructive proof of Pokrzywa's theorem about perturbations of matrix pencils
• Miniversal deformations of chains of linear mappings
• Semi-scalar equivalence of polynomial matrices