Eigenspace of a circulant max-min matrix

Eigenvectors of a max-min matrix characterize stable states of the corresponding discrete-events system. Investigation of the max-min eigenvectors of a given matrix is therefore of a great practical importance. The eigenproblem in max-min algebra has been studied by many authors. Interesting results were found in describing the structure of the eigenspace, and algorithms for computing the maximal eigenvector of a given matrix were suggested, see e.g. [1], [2], [3], [5], [7], [8], [9], [10]. The structure of the eigenspace as a union of intervals of increasing eigenvectors is described in [4]. By max-min algebra we understand a triple (B,⊕,⊗), where B is a linearly ordered set, and ⊕ = max, ⊗ = min are binary operations on B. The notation B(n, n) (B(n)) denotes the set of all square matrices (all vectors) of given dimension n over B. Operations ⊕, ⊗ are extended to matrices and vectors in a formal way. The eigenproblem for a given matrix A ∈ B(n, n) in max-min algebra consists of finding a vector x ∈ B(n) (eigenvector) such that the equation A⊗x = x holds true. By the eigenspace of a given matrix we mean the set of all its eigenvectors. In this paper the eigenspace structure for a special case of so-called circulant matrices is studied. Circulant matrices arise, for example, in applications involving the discrete Fourier transform and the study of cyclic codes for error correction, see [6]. The paper presents a detailed description of all possible types of eigenvectors of any given circulant matrix.