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L-free directed bipartite graphs and echelon-type canonical forms

  • Harm Bart
  • Torsten Ehrhardt
  • Bernd Silbermann
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 271)

Abstract

It is common knowledge that matrices can be brought in echelon form by Gaussian elimination and that the reduced echelon form of a matrix is canonical in the sense that it is unique. In [4], working within the context of the algebra \( \mathbb{C}^{n\times n}_{\mathrm{upper}}\) of upper triangular n×n matrices, certain new canonical forms of echelon-type have been introduced. Subalgebras of \( \mathbb{C}^{n\times n}_{\mathrm{upper}}\) determined by a pattern of zeros have been considered too. The issue there is whether or not those subalgebras are echelon compatible in the sense that the new canonical forms belong to the subalgebras in question. In general they don’t, but affirmative answers were obtained under certain conditions on the given zero pattern. In the present paper these conditions are weakened. Even to the extent that a further relaxation is not possible because the conditions involved are not only sufficient but also necessary. The results are used to study equivalence classes in \( \mathbb{C}^{m\times n}\) associated with zero patterns. The analysis of the pattern of zeros referred to above is done in terms of graph theoretical notions.

Keywords

Echelon (canonical) form zero pattern matrices directed (bipartite) graph partial order L-free graph N-free graph in/out-ultra transitive graph equivalence classes of matrices 

Mathematics Subject Classification (2010)

Primary 15A21 05C50 Secondary 05C20 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Harm Bart
    • 1
  • Torsten Ehrhardt
    • 2
  • Bernd Silbermann
    • 2
    • 3
  1. 1.Econometric Institute, Erasmus University RotterdamRotterdamThe Netherlands
  2. 2.Mathematics DepartmentUniversity of CaliforniaSanta CruzUSA
  3. 3.Fakultät für Mathematik, Technische Universität ChemnitzChemnitzGermany

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