Some Properties Of Circulant Matrices And Applications

In this paper, we mainly discuss nonegative circulant matrices, circulant M-matrices which are closely related to nonegative circulant matrices and the circulant Markov chain-s, a special class of Markov chains. The mainly contents in this paper are as follows:In chapter one, the main concepts and the related theories are introduced, includi-ng the nonegative matrices、the irreducible matrices、the primitive matrix、some related theories, the circulant matrices and its diagnalizations and some general definitions of M-matrices.In chapter two, we investigate norms、genaralized inverses and limits of nonega-tive matrices, where we arrive at the sufficient and necessary conditions for a nonegative matrices to be semiconvergent respectively and get the limit matrices.In chapter three, we discuss the eigenvalues,norms of circulant M-matrices and whether the class of circulant M-matrices or Toplitz M-matrices are closed under some operations.In chapter four, circulant Markov chains are investigated.We reach the sufficient and necessary condition for a nonegative circulant matrix to be irreducible by analysing its eigenvalues and spectral radius; further, we get the final probability distribution matrix of this circulant Markov chain and distinct the states into ergodic sets; last, we get the construction of its accompanied directed graph.