The Bounds Of The Minimum Eigenvalues ??of Two Kinds Of Special Matrices And Their Hadamard Products

Since put forward by the British mathematician Kelly in 19th century, matrix become an important tool of mathematics.It is widely used in statistical analysis, quantum physics, the three dimensional animation production.M-matrix and inverse M-matrix, nonnegative matrix, reducible matrix, irreducible matrix are several types of special matrix. It has important applications in the image analysis, speech processing, robot control. Different from the general matrix, they have special properties. In this paper, we deduce some new propertieson the basis of existing research.M-matrix has a lot of equivalent conditions, we only list 12 equivalent conditions in the paper. Besides,we prove them detailed. Due to the special properties of M-matrix, it has intimate connections with diagonally dominant matrix. we can prove that there is at least one row which is diagonally dominant, it also has a corresponding Hadamard inequalities. similar to positive definite matrix, Schur complement of M-matrix is also a M-matrix. Finally, this paper gives the relevant theorem of minimum eigenvalue of matrix.An inverse M-matrix is the inverse matrix of a M-matrix, a lot of properties can be obtained through the properties of the M-matrices..First we show some common properties, such as every principal sub-matrix of inverse M-matrix is an inverse M-matrix, the principal minor of an inverse M-matrix is greater than zero, P is a permutation matrix, so PT AP is an inverse matrix, and AT is an inverse matrix, for positive diagonal matrix D, E, so DAE is an inverse matrix, then list the equivalent conditions of inverse M-matrix, among them there is the schur complement problem of inverse matrix, and we also prove them in detail.After introducing the properties of M-matrix and inverse M-matrix, this paper studies the Hadamard product of them and its smallest eigenvalue. This problem is a hot issue of research in recent years, many researchers gives better results, on the basis of them, this article gives a further lower bound.