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Canonical matrices for linear matrix problems. (English) Zbl 0967.15007
A large class of matrix problems, which includes the problems of classifying representations of quivers, partially ordered sets and finite dimensional algebras, is studied. Some matrix problems can be formulated in terms of quivers i. e. directed graphs and their representations. The class of studied matrix problems may be extended by considering quivers with relations and partially directed graphs with relations.
The Belitskiĭ’s algorithm [cf. G. R. Belitskiĭ, Normal forms in a space of matrices, in: Marchenko, V. A. (Ed.), Analysis in Infinite-Dimensional Spaces and Operator Theory. Naukova Dumka, Kiev, p. 3-15 (1983) (in Russian)] is presented in a form which allows to reduce pairs of \(n \times n\) matrices to a canonical form by transformations of simultaneous similarity: \((A, B) \mapsto (S^{-1}AS, S^{-1}BS)\). It is shown that the set \(C_{mn}\) of indecomposable canonical \(m \times n\) matrices in the affine space of \(n \times n\) matrices either consists of a finite number of points and straight lines for every \(m \times n\) or that the set \(C_{mn}\) contains a 2-dimensional plane for some \(m \times n\). It means that \(C_{mn}\) satisfies one and only one of the so-called tame and wild type conditions.

MSC:
15A21 Canonical forms, reductions, classification
06A06 Partial orders, general
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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