| The study of idempotent matrices, tripotent matrices, involutive matrices and the idempotency, tripotency, involutory of their linear combinations has important applications in statistical theory, quantum mechanics and control theory. It becomes an active field in theory of matrix algebra, too.Let T1 and T, be n×n nonzero complex matrices, denote a linear combination of the two matrices by T= c1T1+c2T2, where c1, c, are nonzero complex numbers.In Chapter 1, we brief introduce the significance and the research state at home and abroad about the problem of the linear combinations of matrices. In Chapter 2, the basic knowledge on matrix theory is introduced. In Chapter 3, the main results:We give all forms of the arbitrary matrix T2 for T is an involutive matrix or s+1-potent matrix, respectively, where T1 is a tripotent matrix, an anti-idempotent matrix, or involutive matrix, respectively, with T1T2= T2T1. In Chapter 4, we give the sufficient and necessary conditions for T is a t+1-potent matrix, where T1 is a tripotent matrix and an idempotent matrix, respectively, T2 is a s+1-potent matrix and commutative withl T1. Study of the paper mostly makes use of similar normalized form of tripotent matrices, anti-idempotent matrices, and involutive matrices.The main results are listed below:1. Let T1 be an anti-idempotent matrix with n-order, T2 be an arbitrary matrix of the same order with T1. Then the characteristics of the forms of T2 are given, under T1T2= T2T1 condition, in which the linear combination c1T1+c2T2(c1,c2∈C\{0}) be involutive.2. Let T1 be an involutive matrix with n-order, T2 be an arbitrary matrix of the same order with T1. Then the characteristics of the forms of T, are given, under T1T2= T2T1 condition, in which the linear combination c1T1+c2T2 (c1, c2 E C\{0}) be involutive.3. Let T1 be a tripotent matrix with n-order, T2 be an arbitrary matrix of the same order with T1. Then the characteristics of the forms of T2 are given, under T1T2= T2T1 condition, in which the linear combination c1T1+c2T2 (c1,c2∈\{0}) be s+1-potent.4. Let T1 be an anti-idempotent matrix with n-order, T2 be an arbitrary matrix of the same order with T1. Then the characteristics of the forms of T2 are given, under T1T2= T2T1 condition, in which the linear combination c1T1+c2T2(c1,c2∈C\{0}) be s+1-potent. 5.Let T1 be an involutive matrix with n-order, T2 be an arbitrary matrix of the same order with T1.Then the characteristics of the forms of T2 are given,under T1T2=T2T1 condition,in which the linear combination c1T1+c2T2(c1,c2∈C\{0})be s+1-potent.6.Let T1 be a tripotent matrix with n -order, T2 be a s+1-potent matrix of the same order with T1.Then giving the sufficient and necessary conditions of the linear combination c1T1+c2T2(c1,c2∈C\{0})is t+1-potent,under T1T2=T2T1 condition.7.Let T1 be an idempotent matrix with n -order,T2 be a s+1-potent matrix of the same order with T1. Then giving the sufficient and necessary conditions of the linear combination c1T1+c2T2(c1,c2∈C\{0})is t+1-potent,under T1T2=T2T1 condition. |
