On commuting matrices in max algebra and in classical nonnegative algebra

On Commuting Matrices in Max Algebra and in Classical Nonnegative Algebra

This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we ...

Nonnegative matrices, max-algebra and applications

The Algebra Generated by Three Commuting Matrices

We present a survey of an open problem concerning the dimension of the algebra generated by three commuting matrices. This article concerns a problem in algebra that is completely elementary to state, yet, has proven tantalizingly difficult and is as yet unsolved. Consider C[A,B,C] , the C-subalgebra of the n × n matrices Mn(C) generated by three commuting matrices A, B, and C. Thus, C[A,B,C] c...

On Semigroups of Matrices in the (max,+) Algebra

We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max;+)-algebra (that is, the semiring (R[ f 1g;max;+)), and also for semigroups of (max;+)-linear projective maps with rational entries. An application to the estimation of the Lyapunov exponent of certain products of random matrices is also discussed. Key-words: Semigroups, Burnside Prob...

Max-Plus algebra on tensors and its properties

In this paper we generalize the max plus algebra system of real matrices to the class of real tensors and derive its fundamental properties. Also we give some basic properties for the left (right) inverse, under the new system. The existence of order 2 left (right) inverses of tensors is characterized.