The Research On Tripotency And Involutorial Of Linear Combinations Of Some Matrices

As many problems in system theory, statistics, and cryptography are equivalent to the problems of linear combinations of matrices, for example, in singular system, the regularity of singular system is equivalent to the nonsingularity of linear combination of two matrices. Hence, more people pay attention to the research on the linear combinations of the matrices. And the research on some special matrices as idempotent matrix, tripotent matrix, involutive matrix and the properties of their linear combinations have important significance in matrix theory and many applications field.In Chapter 1, the research significance and research situation of this subject are briefly introduced, and in Chapter 2, the basic knowledge related to this paper is introduced, the main results of this paper in Chapter 3 and 4 are listed below:(1) Let A be an idempotent matrix of order n and B be an t-potent matrix of order n, which satisfy the condition of AB=BA. Then the sufficient and necessary conditions for the linear combination aA+bB is a tripotent matrix are given, where (a,b∈C\{0});(2) Let A be a tripotent matrix of order n and B be an involutive matrix of order n. Then the sufficient and necessary conditions for the linear combination aA+bB is an involutive matrix are given, where a, b∈C\{0};(3) Let A and B be tripotent matrices of order n, which satisfy the condition A2B= BA2. Then the sufficient and necessary conditions for the linear combination aA+bB is an involutive matrix are given, where a, b∈C\{0};(4)) Let A be an idempotent matrix of order n, and B be an tripotent matrix of order n. Then the characteristics of all cases in which the linear combination aA+bB is an involutive matrix are given, where a, b∈C\{0}.