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Abstract
We study the generalized eigenproblem A ⊗ x = λ ⊗ B ⊗ x, where A, B ∈ R^{m×n} in the maxplus 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.
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 language  English 

Pages (fromto)  1002–1021 
Number of pages  20 
Journal  S I A M Journal on Matrix Analysis and Applications 
Volume  37 
Issue number  3 
Early online date  4 Aug 2016 
DOIs  
Publication status  Epub ahead of print  4 Aug 2016 
Keywords
 Matrix
 Maxplus Algebra
 Generalized Eigenproblem
 Spectrum
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Dive into the research topics of 'On special cases of the generalized maxplus eigenproblem'. Together they form a unique fingerprint.Projects
 1 Finished

PerronFrobenius Theory and MaxAlgebraic Combinatorics of Nonnegative Matrices
Butkovic, P.
Engineering & Physical Science Research Council
12/03/12 → 11/03/14
Project: Research Councils