Research On Some Problems In Matrix Space

The discussion of the relationship between quantities occupies an important place in the field of basic mathematics.The relationship between quantities can be divided into two types.One is the study of relationships that preserve certain invariants on a given mathematical structure,while in matrix analysis,studying the mapping structure of preserving invariants between matrix sets is called the preserver problems of matrices;the other relationship is the study of some variables in a specific non-empty set.In matrix theory,the study of variables between matrices is called matrix inequality problem.The problem of matrix preservation and inequality has become the core research problem in matrix theory and has a wide range of practical applications in control system theory,operational research and other fields.The paper mainly studies the relationship between matrix and tensor(hypermatrix),which can be divided into the following aspects.In chapter one,the relationship between matrices and tensor base on Einstein product is investigated.All linear isomorphism between tensor spaces and matrix spaces preserving Einstein product are obtained.The result generalizes transformation given by Brazell et al.In chapter two,firstly,it generalizes some matrix basic inequalities to tensors;secondly,it combines the arithmetic-geometric mean inequality and the H(?)lder inequality about unitary invariant norm to obtain a unitary invariant norm inequality.In this paper,a majorization relation on the positive semi-definite block matrix is established.The result considers the sum of multiple matrices and controls the sum symbol,which can bring better matrix perturbation effect and can be better used in control engineering and other fields.The proof has important implications for the development of matrix theory.