Normal Tropical (0,−1)-Matrices and Their Orthogonal Sets

B. Bakhadly, A. Guterman, M. J. de la Puente

Research output: Contribution to journalArticlepeer-review

Abstract

Square matrices A and B are orthogonal if A ⨀ B = Z = B ⨀ A, where Z is the matrix with all entries equal to 0, and ⨀ is the tropical matrix multiplication. We study orthogonality for normal matrices over the set {0,−1}, endowed with tropical addition and multiplication. To do this, we investigate the orthogonal set of a matrix A, i.e., the set of all matrices orthogonal to A. In particular, we study the family of minimal elements inside the orthogonal set, called a basis. Orthogonal sets and bases are computed for various matrices and matrix sets. Matrices whose bases are singletons are characterized. Orthogonality and minimal orthogonality are described in the language of graphs. The geometric interpretation of the results obtained is discussed.

Original languageEnglish
Pages (from-to)614-631
Number of pages18
JournalJournal of Mathematical Sciences
Volume269
Issue number5
DOIs
StatePublished - Feb 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Statistics and Probability
  • General Mathematics

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