Structures Of Circulant Inverse M-Matrices

Inverse M-matrices and circulant matrices are two classes of important matrices. Inverse M-matrices often occur in relation to systems of linear or non-linear equations or eigenvalues problems in a wide variety of areas including finite difference methods for partial differential equations, input-output production and growth model in economics, iterative methods in numerical analysis, and Markov processes in probability and statistics. Circulant matrices are often used as preconditioner for Toeplitz linear systems since they can be easily inverted and super-fast computed.In this paper, an interesting result on the structures of circulant inverse M-matrices is presented.It is shown that if the n×n nonnegative circulant matrix is not a positive matrix and not equal to c₀ I, then A is an inverse M-matrix if and only if there exists a positive integer k, which is a proper factor of n, such that c_jk>0 for j = 0,1,…, [ (n-k)/k], the others c_i are zero and Circ[ c₀ ,c_k ,...,c_n-k] is an inverse M-matrix. The result is then extended to so-called generalized circulant inverse M-matrices.