On the Sequence of Consecutive Powers of a Matrix in a Boolean Algebra

If you want to cite this report, please use the following reference instead: B. De Schutter and B. De Moor, " On the sequence of consecutive powers of a matrix in a Boolean algebra, " SIAM Abstract. In this paper we consider the sequence of consecutive powers of a matrix in a Boolean algebra. We characterize the ultimate behavior of this sequence, we study the transient part of the sequence and we derive upper bounds for the length of this transient part. We also indicate how these results can be used in the analysis of Markov chains and in max-plus-algebraic system theory for discrete event systems. 1. Introduction. In this paper we consider the sequence of consecutive powers of a matrix in a Boolean algebra. This sequence reaches a " cyclic " behavior after a finite number of terms. Even for more complex algebraic structures, such as the max-plus algebra (which has maximization and addition as its basic operations) this ultimate behavior has already been studied extensively by several authors (See, e.g., [1, 9, 13, 26] and the references therein). In this paper we completely characterize the ultimate behavior of the sequence of the consecutive powers of a matrix in a Boolean algebra. Furthermore, we also study the transient part of this sequence. More specifically, we give upper bounds for the length of the transient part of the sequence as a function of structural parameters of the matrix. Our main motivation for studying this problem lies in the max-plus-algebraic system theory for discrete event systems. Furthermore, our results can also be used in the analysis of the transient behavior of Markov chains. This paper is organized as follows. In §2 we introduce some of the notations and concepts from number theory, Boolean algebra, matrix algebra and graph theory that will be used in the paper. In §3 we characterize the ultimate behavior of the sequence of consecutive powers of a given matrix in a Boolean algebra, and we derive upper bounds for the length of the transient part of this sequence. In §4 we briefly sketch how our results can be used in the analysis of Markov chains and in the max-plus-algebraic system theory for discrete event systems. In this section we also explain why we have restricted ourselves to Boolean algebras in this paper and we indicate some of the phenomena that should be taken into account when extending …