The Preserver Of Magic Integer Matrix And JAD Conjecture

As an important branch of algebra,matrix theory plays a very important role in every research field of algebra,many branches of mathematics,computer graphics and other field.The integer matrix named for its element's own characteristics has also attracted much attention,people have studied it from the aspects of the generalized inverse,the solution to the integer matrix equation and its application of the cryptosystem,the researches results are quite rich.The problem of matrix preserver has become a hot research topic in matrix theory since it was put forward,the research results are increasingly rich,on the one hand,it has important theoretical value;on the other hand,it has wide application prospects in mathematical statistics,quantum mechanics,differential geometry and other fields.In this paper,we will study the magic integer matrix and its preserver from the following three aspects.Firstly,using the definition of Hadamard product of matrices and the related properties of integer division,the definition of the integer matrix's Hadamard division is given,and then the unique decomposition theorem of the integer matrix's Hadamard product is obtained.Secondly,the definition and simple properties of magic integer matrix are given by Hadamard product of matrices.And then the linear preserver of magic matrices and the magic-sum preserver of magic squares are studied.Finally,The conjecture of magic square(JAD conjecture)proposed by J.Dénes and A.D.kecdWell in the article entitled "A Conjecture Concerning Magic" published in American Mathematical Monthly in 1988 is studied,and the proof of JAD conjecture is completed for the first time.