On special cases of the generalized max-plus eigenproblem

Peter Butkovic, Daniel Jones

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5 Citations (Scopus)
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We study the generalized eigenproblem A ⊗ x = λ ⊗ B ⊗ x, where A, B ∈ Rm×n in the max-plus algebra.  It is known that if A and B are symmetric, then there is at most one generalized eigenvalue, but no description of this unique candidate is known in general. We prove that  if C = A − B is symmetric, then the common value of all saddle points of C (if any) is the
unique candidate for λ. We also explicitly describe the whole spectrum in the case when B is an outer product. It follows that when A is symmetric and B is constant, the smallest column maximum of A is the unique candidate for λ. Finally, we provide a complete description of the spectrum when n = 2.
Original languageEnglish
Pages (from-to)1002–1021
Number of pages20
JournalS I A M Journal on Matrix Analysis and Applications
Issue number3
Early online date4 Aug 2016
Publication statusE-pub ahead of print - 4 Aug 2016


  • Matrix
  • Max-plus Algebra
  • Generalized Eigenproblem
  • Spectrum


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