| In this paper, we study a class of important and special structure matrix --- the circluant M-matrix,which are an important component of the matrix theory. In addtion , circluant M-matrix have become one of the most important and active research fields of applied mathematics increasingly. Because of many good properties and structure of circulant matrix, it is necessary to study their special properties,structure,the inverse eigenvalue proplem. the inverse problem of the matrix is considered : given the limited conditions ,find the matrix with the prescribed eigenvalue or eigenvetor.The content of the paper is arranged as follows:Chapter one, we carry on the introduction and prearrangement .In this part,we present the introduction of the research direction and the trends of circulant matrix ,we also offer prearrangement.Chapter two, we study the properties of the symmetric circluant M-matrix as follows: the range of eigenvalue and the existence of the inverse matrix and we discuss the properties of limit and 2-norm for the inverse matrix -degree k.In the end , the perurbation theorem of the invese matrix is presented.Chapter three , we give the inverse eigenvalue problem of the circulant M-matrix with prescribed real spectrum and complex spectrum, respectively.In fact ,we can assure that the real circulant M-matrix of real spectrum is symmetric.Meanwhile, the circulant M-matrix of complex spectrum is dissymmetric.So we discuss in deed the inverse eignvalue problem of symmetric and dissymmetric circulant M-matrix .Besides,we present the correspond numerical examples .Chapter four, based on the charpter three, we discuss farther the inverse eignvalue problem of the circulant inverse M- matrix. Similarily, we construct the circulant inverse M-matrix with the prescribed real spectrum and complex spectrum, we also offer the correspond numerical examples .In this paper ,we use science software–Matlab 6.5 to testify the right of all numerical examples.
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