Unitary matrix digraphs and minimum semidefinite rank

Yunjiang Jiang, Lon H. Mitchell, Sivaram K. Narayan

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

For an undirected simple graph G, the minimum rank among all positive semidefinite matrices with graph G is called the minimum semidefinite rank (msr) of G. In this paper, we show that the msr of a given graph may be determined from the msr of a related bipartite graph. Finding the msr of a given bipartite graph is then shown to be equivalent to determining which digraphs encode the zero/nonzero pattern of a unitary matrix. We provide an algorithm to construct unitary matrices with a certain pattern, and use previous results to give a lower bound for the msr of certain bipartite graphs.

Original languageEnglish
Pages (from-to)1685-1695
Number of pages11
JournalLinear Algebra and Its Applications
Volume428
Issue number7
DOIs
StatePublished - Apr 1 2008

Keywords

  • Digraph
  • Graph
  • Positive semidefinite
  • Quadrangular
  • Rank
  • Unitary

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