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On the existence of matrices with prescribed height and level characteristics

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Abstract

We determine all possible relations between the height (Weyr) characteristic and the level characteristic of anM-matrix. Under the assumption that the two characteristics have the same number of elements, we determine the possible relations between the two characteristics for a wider class of matrices, which also contains the class of strictly triangular matrices over an arbitrary field. Given two sequences which satisfy the above condition, we construct a loopless acyclic graphG with the following property: Every matrix whose graph isG has its height characteristic equal to the first sequence and its level characteristic equal to the second. We give several counterexamples to possible extensions of our results, and we raise some open problems.

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Authors and Affiliations

  1. Mathematics Department, Technion, 32000, Haifa, Israel

    Daniel Hershkowitz

  2. Mathematics Department, University of Wisconsin-Madison, 53706, Madison, WI, USA

    Hans Schneider

Authors
  1. Daniel Hershkowitz
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  2. Hans Schneider
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Additional information

The research was carried out while the author was a visiting professor at the University of Wisconsin — Madison

The research of this author was supported in part by NFS grants DMS-8521521, DMS-8901445, and EMS-8718971, and by the International Scientific Research Program, the Ministry of Education, Science and Culture, Japan.

The research of both authors was supported also by their joint grant No. 90-00434 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.

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Hershkowitz, D., Schneider, H. On the existence of matrices with prescribed height and level characteristics. Israel J. Math. 75, 105–117 (1991). https://doi.org/10.1007/BF02787184

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  • DOI: https://doi.org/10.1007/BF02787184

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