The Special Matrix Such As M-matrix And The Special Product

Special matrix is the matrix that it's entry in value or it's property is characteristic. From the big aspect, they may be divided into two parts. A part is depicted by containing other nature that is difficult to directly perceived. We call that special property matrix or nature matrix; Another part is depicted by containing other nature that is easy to directly perceived. We call that special type matrix. Special matrix has the value of its self and play a unique role no matter on science or application. On one hand, there is certain application background in most matrix types: On the other hand, can again draw forth the type of some matrices from the research of application program.This paper discusses and researches some special matrix, and mainly does following several aspects.First, The special products of some matrices, as Fan product, Kronecker product and Hadamard product, are discussed. Some little results such as the Kronecker product of normal matrix is still normal matrix are obtained. The relation between the theory of M-matrix and the study of economics is introduced briefly.Second, M-matrix, inverse M-matrix and H-matrix are discussed in varying degree, and the minimal eigenvalue of M-matrix is discussed especially. There are the main results: If A Rnxn is M- matrix and r is odd number, then lr(A) l(Aor); If A Rnxn is M-matrix and A is any principal submatrix of A, then l(A) l(A): If A, B eRn n are irreducible M-matrices, then there exist diagonal matrices D1 =diag(d1, ..., dn) and D2 =diag(d1, ..., dn) with positive diagonalentries such that D1A-1D2 is doubly stochastic and l(A o B-1) > min dkdkbkk,where B-1 = [bij]. On the base of the conclusion, a result that has already existed (if A Rn n is irreducible M-matrix, then there exist diagonal matrices D1 and D2 with positive diagonal entries such that D1A-1D2 is doubly stochastic.) is improved, i.e. if A Rnxn is irreducible M-matrix and A-1 = [aij], then there exist diagonal matrices D1 =diag(d1, ..., dn) D2 =diag(d1, ..., dn] with positive diagonal entries such that D1A-1D2 is doublv stochastic and min dk min dk < max dk max dk l(A). While the inverse M-matrix is dis-cussed, the equivalent condition for the sum of two inverse M-matrices is still inverse M-matrix is obtained; The result that if A = [aij] Rnxn is inverse M-matrix and diagonally dominant of its rows and B = [bij] eRnxn is inverse M-matrix, then A*B is M-matrix is gotten, and some results such as the Schur complement of inverse M-matrix is still inverse M- matrix are reached. In this paper, some fundamental property of H-matrix are studied, and popularize the definition of the minimal eigenvalue on Z-matrix to //-matrix, and some results such as if A, B 拢 C"xn are H- matrices, then i(AoB) = l(A*B] > I (A) 鈥?l(B] which are similar to the minimal eigenvalue of M-matrix are gotten.Third, the various conditions about M-matrix which Plemmons, Neumann, etc. sum up are weaken, and put forward the concept of generalized M- matrix. A kind of generalized M-matrix is discussed, and some equivalent conditions relating the definition of this kind of generalized M-matrix are obtained.