The Special Product And Multiplicative Commutators

Matrix theory is an important branch of algebra, and also a kind of basic mathe-matical tools, and have important application areas in mathematics and many other disciplines. At present, matrix theory has been widely used in wireless communi-cations, financial statistics, system engineering, optimization theory, the electronic simulation and other fields of engineering, especially in the image encryption and digital water Printing applications and the effective combination of Matlab. In this thesis, we mainly put forward some special matrices do special products on the basis of forefathers’research. The construction of chapters and the concrete contents of this paper are as follows:In the first chapter, about Hadamard products of Hadamard powers of nonneg-ative matrices are studied. For general nonnegative matrix, the spectral radius is not a convex function of its nonnegative matrix. However, the Logarithm of the spectral radius of Hadamard products of Hadamard powers of nonnegative matrices, is a convex function from R+k to R. Secondly, some sufficient and necessary con-ditions about Hadamard products of Hadamard powers of irreducible nonnegative matrices is given.In the second chapter, some application of multiplicative commutators are giv-en. The sufficient and necessary conditions to write A=X1X2X3, B=XT1XT2XT3for some nonsingular Xi∈Mn(i=1,2,3), where т∈S3is given.In the third chapter, the Hadamard product and Fan product of M-matrix are reserched. This chapter on the basis of the lemma(3.1.6), which has been generalized to the general complex matrix. Secondly, on the basis of the lemma (3.1.7), we got about a lower bound of matrix inequality between the Fan product and the Hadamard product.In the fourth chapter, some related inequalities of Hadamard product are re-serched. First of all, The theorem(4.1.8) will been generalized to the form of p-fold Hadamard product; Secondly, we applied positive semidefinite partial order to the-orem(5.3.1) given in literature [1] and provide another kind of proof of it; Again, according to the definition (4.1.5), we get some related properties of Φ(A)and Φ Finally, on the basis of Perron-Frobenius theorem and the ideas of literature [1,2], some related inequalities of Hadamard product are obtained.